BiomedischeFysicaLO_1sem.pdf

# Introduction to mechanics and biomechanics Mechanics is the study of forces and their effects, and biomechanics applies these principles to biological systems [5](#page=5). ### 1.1 Core concepts of mechanics Mechanics is divided into two main branches: * **Statics**: The study of bodies at rest or in equilibrium due to acting forces [5](#page=5) [8](#page=8). * **Dynamics**: The study of moving bodies [5](#page=5). Dynamics is further subdivided into: * **Kinematics**: The study of the properties of motion without considering the physical causes. It focuses on the geometric aspects of movement and establishes relationships between vector quantities such as position, velocity, and acceleration [5](#page=5) [6](#page=6). * **Kinetics**: The study of the forces responsible for motion. Kinetic analyses consider factors like gravity, friction, air or water resistance, and muscle contractions. These analyses often utilize Newton's second law in various mathematical forms [5](#page=5) [7](#page=7). > **Tip:** To conduct a thorough analysis of any motion, both kinematic and kinetic parameters must be investigated [7](#page=7). ### 1.2 Types of motion To analyze movement systematically, motion is categorized into linear, angular, or general types [5](#page=5). #### 1.2.1 Linear motion (translation) In linear motion, all parts of a body move the same distance in the same amount of time and in the same direction. Figure 1 provides examples of linear motion, such as the path of a gymnast's center of mass or a baseball [5](#page=5) [6](#page=6). #### 1.2.2 Angular motion (rotation) Angular motion, or rotation, occurs when all parts of a body move through the same angle in a given time interval. Rotational movements happen around an axis of rotation, which can be internal to the system, pass through the center of mass, or be external. Figure 2 illustrates rotational motion around a shoulder joint, the center of mass, or an external axis [5](#page=5) [6](#page=6). #### 1.2.3 General motion General motion occurs when translation and rotation happen simultaneously [5](#page=5). ### 1.3 Application in biomechanics Biomechanics applies mechanical principles to understand biological systems, such as human and animal movement or posture. A biomechanical analysis may involve [5](#page=5): * **Kinematic analysis**: Describing movement characteristics without reference to the forces causing it. For example, analyzing the linear motion of the entire body coupled with the angular motion of body parts during activities like walking [6](#page=6) [7](#page=7) [8](#page=8). * **Kinetic analysis**: Examining the forces responsible for motion. This can involve analyzing vertical ground forces during weightlifting to understand lifting movements and the moments of force in joints related to muscle activation [7](#page=7). > **Example:** A dynamic analysis of walking would include a kinematic component describing the whole body's linear motion and body parts' angular motion, linked to a kinetic analysis of the ground and joint forces required for the gait [7](#page=7) [8](#page=8). While sophisticated biomechanical situations may require advanced mathematical calculations, the fundamental approach remains consistent with the basic principles of mechanics [8](#page=8). --- # Linear kinematics: motion analysis This section introduces the fundamental principles of linear kinematics, focusing on how to describe and analyze motion through concepts like reference frames, position, displacement, velocity, and acceleration [9](#page=9). ### 2.1 Reference systems and position Movement is relative, so the first step in describing motion is choosing a reference system. This system consists of an origin and a set of axes (one-dimensional, two-dimensional, or three-dimensional). In three dimensions, an orthogonal, right-handed coordinate system with x, y, and z axes is typically used [10](#page=10) [9](#page=9). A point's location can be specified by its coordinates (x, y, z), which are the lengths of the perpendicular projections of a line segment from the origin to the point onto each axis. Alternatively, a position vector, denoted by $\mathbf{r}$, can represent the location of a point P with an arrow drawn from the origin O to P. A vector possesses magnitude, direction, and sense [10](#page=10). The change in position from point A to point B is described by the displacement vector $\vec{AB}$. The path, or trajectory, is the collection of points a body traces during its motion. It's important to note that the displacement does not necessarily coincide with the path [11](#page=11). ### 2.2 Vectors Physical quantities can be represented as either scalar or vector quantities [11](#page=11). #### 2.2.1 Scalar quantities A scalar quantity is a number with an associated unit. It can be positive, negative, or zero. Standard algebraic rules apply to scalar quantities. Examples include length, mass, and temperature [11](#page=11). #### 2.2.2 Vector quantities A vector quantity has magnitude, direction, and sense. Vectors are graphically represented by arrows where the length is proportional to the magnitude. Standard algebraic rules do not apply to vector operations in the same way as scalars [11](#page=11). > **Tip:** Remember that vector addition is not simply the sum of their magnitudes. For instance, the displacement from A to B plus the displacement from B to C equals the displacement from A to C ($\vec{AB} + \vec{BC} = \vec{AC}$), but the sum of the lengths of $\vec{AB}$ and $\vec{BC}$ is not necessarily equal to the length of $\vec{AC}$ [11](#page=11). #### 2.2.3 Components of a vector The components of a vector are its (perpendicular) projections onto the axes of a coordinate system. These components are oriented line segments with signs, making them scalar quantities [12](#page=12). In two dimensions, for a vector $\mathbf{a}$ with components $a_x$ and $a_y$ along the x and y axes respectively, and $\phi$ as the angle with the x-axis: * $a_x = a \cos \phi$ [12](#page=12). * $a_y = a \sin \phi$ [12](#page=12). * The magnitude is $a = \sqrt{a_x^2 + a_y^2}$ [12](#page=12). * The vector can be expressed as $\mathbf{a} = a_x \mathbf{i} + a_y \mathbf{j}$, where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors along the x and y axes [12](#page=12). In three dimensions, for a vector $\mathbf{a}$ with components $a_x$, $a_y$, and $a_z$ along the x, y, and z axes, respectively, and unit vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$: * $\mathbf{a} = a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k}$ [12](#page=12). * The magnitude is $a = \sqrt{a_x^2 + a_y^2 + a_z^2}$ [12](#page=12). #### 2.2.4 Sum and difference of vectors Vectors can be added or subtracted graphically or using their components [13](#page=13). The graphical method for vector addition involves placing the tail of the second vector at the head of the first. The resultant vector is drawn from the tail of the first vector to the head of the second. The difference $\mathbf{a} - \mathbf{b}$ is equivalent to $\mathbf{a} + (-\mathbf{b})$ [13](#page=13). When working with components: * If $\mathbf{c} = \mathbf{a} + \mathbf{b}$, then $c_x = a_x + b_x$ and $c_y = a_y + b_y$ [13](#page=13). * If $\mathbf{c} = \mathbf{a} - \mathbf{b}$, then $c_x = a_x - b_x$ and $c_y = a_y - b_y$ [13](#page=13). #### 2.2.5 Product of vectors There are three vector product operations with physical significance: scalar multiplication of a vector, the scalar product (dot product) of two vectors, and the vector product (cross product) of two vectors [14](#page=14). ##### 2.2.5.1 Product of a scalar and a vector Multiplying a scalar $k$ by a vector $\mathbf{a}$ results in a vector $k\mathbf{a}$ with magnitude $|k|a$ and the same direction as $\mathbf{a}$ if $k$ is positive, and the opposite direction if $k$ is negative [14](#page=14). ##### 2.2.5.2 Scalar product of vectors The scalar product of vectors $\mathbf{a}$ and $\mathbf{b}$ is defined as: $$ \mathbf{a} \cdot \mathbf{b} = ab \cos \phi $$ [15](#page=15). where $\phi$ is the angle between the vectors. The result is a scalar quantity [15](#page=15). Geometrically, the scalar product represents the product of the projection of one vector onto the other, multiplied by the magnitude of the second vector [15](#page=15). * If $\mathbf{a}$ and $\mathbf{b}$ are parallel, $\mathbf{a} \cdot \mathbf{b} = ab$ [15](#page=15). * If $\mathbf{a}$ is perpendicular to $\mathbf{b}$ ($\phi = 90^\circ$), $\mathbf{a} \cdot \mathbf{b} = 0$ [15](#page=15). Analytically, for vectors with components: $$ \mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y + a_z b_z $$ [15](#page=15). Properties of the scalar product: * Commutativity: $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$ [15](#page=15). * Distributivity over addition: $(\mathbf{a} + \mathbf{b}) \cdot \mathbf{c} = \mathbf{a} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{c}$ [15](#page=15). * Scalar multiplication: $k(\mathbf{a} \cdot \mathbf{b}) = (k\mathbf{a}) \cdot \mathbf{b} = \mathbf{a} \cdot (k\mathbf{b})$ [15](#page=15). ##### 2.2.5.3 Vector product of vectors The vector product of vectors $\mathbf{a}$ and $\mathbf{b}$ is defined as: $$ \mathbf{c} = \mathbf{a} \times \mathbf{b} $$ [15](#page=15). The result is a new vector $\mathbf{c}$ with specific properties: * Magnitude: $c = ab \sin \phi$, where $\phi$ is the smallest angle from $\mathbf{a}$ to $\mathbf{b}$ [15](#page=15). * Direction: $\mathbf{c}$ is perpendicular to the plane formed by $\mathbf{a}$ and $\mathbf{b}$ [15](#page=15). * Sense: The sense of $\mathbf{c}$ is determined by the right-hand rule (corkscrew rule) when rotating from $\mathbf{a}$ to $\mathbf{b}$ [15](#page=15). * If $\mathbf{a}$ and $\mathbf{b}$ are parallel, $\mathbf{a} \times \mathbf{b} = \mathbf{0}$ [16](#page=16). * If $\mathbf{a}$ is perpendicular to $\mathbf{b}$, $|\mathbf{a} \times \mathbf{b}| = ab$ [16](#page=16). Analytically, the vector product is given by: $$ \mathbf{a} \times \mathbf{b} = (a_y b_z - a_z b_y)\mathbf{i} + (a_z b_x - a_x b_z)\mathbf{j} + (a_x b_y - a_y b_x)\mathbf{k} $$ [16](#page=16). Properties of the vector product: * Anti-commutativity: $\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})$ [16](#page=16). * Distributivity over addition: $\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) + (\mathbf{a} \times \mathbf{c})$ [16](#page=16). ### 2.3 Velocity Velocity describes the rate of change of an object's position over time [17](#page=17). #### 2.3.1 Average velocity The average velocity ($\mathbf{v}_{avg}$) over a time interval $(\Delta t)$ is defined as the displacement ($\Delta \mathbf{r}$) divided by the time interval: $$ \mathbf{v}_{avg} = \frac{\Delta \mathbf{r}}{\Delta t} $$ [17](#page=17). Average velocity is a vector quantity. If the object returns to its starting point (a closed trajectory), the displacement is zero, and thus the average velocity is zero. It's important to distinguish this from average speed, which considers the total distance traveled [17](#page=17) [18](#page=18). > **Example:** An athlete sprints 50 m in 8 s and then walks back to the start in 40 s. > * Average sprint velocity: $\Delta x = 50$ m, $\Delta t = 8$ s, so $\mathbf{v}_{avg, sprint} = \frac{50 \text{ m}}{8 \text{ s}} = 6.25$ m/s [17](#page=17). > * Average walking velocity: $\Delta x = -50$ m, $\Delta t = 40$ s, so $\mathbf{v}_{avg, walk} = \frac{-50 \text{ m}}{40 \text{ s}} = -1.25$ m/s [17](#page=17). > * Average velocity for the entire motion: $\Delta x = 0$ m, $\Delta t = 48$ s, so $\mathbf{v}_{avg, total} = \frac{0 \text{ m}}{48 \text{ s}} = 0$ m/s [18](#page=18). > * Average speed for the entire motion: Total distance = 100 m, $\Delta t = 48$ s, so average speed = $\frac{100 \text{ m}}{48 \text{ s}} \approx 2.08$ m/s [18](#page=18). #### 2.3.2 Instantaneous velocity Instantaneous velocity ($\mathbf{v}$) is the velocity of an object at a specific moment in time. It is found by taking the limit of the average velocity as the time interval approaches zero: $$ \mathbf{v} = \lim_{\Delta t \to 0} \frac{\Delta \mathbf{r}}{\Delta t} = \frac{d\mathbf{r}}{dt} $$ [18](#page=18). Thus, instantaneous velocity is the derivative of the position vector with respect to time. It is a vector with a magnitude $v = |\frac{d\mathbf{r}}{dt}|$ and its direction is tangent to the path of motion [19](#page=19). The dimensions of velocity are length per time ($l t^{-1}$), and its SI unit is meters per second (m/s) [19](#page=19). #### 2.3.3 Graphical interpretation of velocity In a position-time (t,x) graph: * The average velocity between two points is the slope of the straight line connecting those points [20](#page=20). * The instantaneous velocity at a specific point is the slope of the tangent line to the curve at that point. A positive slope indicates increasing position with time (positive velocity), while a negative slope indicates decreasing position with time (negative velocity) [20](#page=20). > **Tip:** A steeper slope on a (t,x) graph indicates a higher velocity [20](#page=20). ### 2.4 Acceleration Acceleration is the rate at which an object's velocity changes over time [21](#page=21). #### 2.4.1 Average acceleration The average acceleration ($\mathbf{a}_{avg}$) over a time interval $(\Delta t)$ is the change in velocity ($\Delta \mathbf{v}$) divided by the time interval: $$ \mathbf{a}_{avg} = \frac{\Delta \mathbf{v}}{\Delta t} = \frac{\mathbf{v}_2 - \mathbf{v}_1}{t_2 - t_1} $$ [22](#page=22). #### 2.4.2 Instantaneous acceleration Instantaneous acceleration ($\mathbf{a}$) is the acceleration at a specific moment in time. It is found by taking the limit of the average acceleration as the time interval approaches zero: $$ \mathbf{a} = \lim_{\Delta t \to 0} \frac{\Delta \mathbf{v}}{\Delta t} = \frac{d\mathbf{v}}{dt} $$ [22](#page=22). Acceleration is the derivative of the velocity vector with respect to time. It is a vector with magnitude $a = |\frac{d\mathbf{v}}{dt}|$. Its direction is tangent to the "hodograph," which is the path traced by the velocity vector originating from a common point [22](#page=22). The dimensions of acceleration are length per time squared ($l t^{-2}$), and its SI unit is meters per second squared (m/s²) [23](#page=23). * **Speeding up:** If the acceleration vector is parallel or anti-parallel to the velocity vector, the magnitude of the velocity changes. If they have the same sign (e.g., both positive or both negative in 1D), the object speeds up [23](#page=23). * **Slowing down:** If the velocity and acceleration have opposite signs, the object slows down (decelerates) [23](#page=23). * **Changing direction:** If the acceleration vector is perpendicular to the velocity vector, the magnitude of the velocity remains constant, but its direction changes [23](#page=23). > **Tip:** A positive acceleration does not always mean speeding up, and a negative acceleration does not always mean slowing down. The sign of the acceleration relative to the sign of the velocity determines whether the object speeds up or slows down in one-dimensional motion [23](#page=23). | Velocity (v) | Acceleration (a) | Outcome | | :----------- | :--------------- | :------------------------------------ | | + | + | Speeding up in the positive direction | | - | - | Speeding up in the negative direction | | + | - | Slowing down | | - | + | Slowing down | | ± | 0 | Constant velocity | #### 2.4.3 Graphical interpretation of acceleration In a velocity-time (t,v) graph: * The average acceleration between two points is the slope of the straight line connecting those points [25](#page=25). * The instantaneous acceleration at a specific point is the slope of the tangent line to the curve at that point [25](#page=25). ### 2.5 Relationship between position, velocity, and acceleration In one-dimensional motion, these quantities are related through differentiation and integration: * Given position $x(t)$, velocity is $v(t) = \frac{dx}{dt}$ and acceleration is $a(t) = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ [26](#page=26). * Given velocity $v(t)$, position can be found by integration: $x(t) = x_0 + \int_{0}^{t} v(t') dt'$ [26](#page=26). * Given acceleration $a(t)$, velocity can be found by integration: $v(t) = v_0 + \int_{0}^{t} a(t') dt'$. Position can then be found by integrating the velocity: $x(t) = x_0 + \int_{0}^{t} v(t') dt'$ [27](#page=27). #### 2.5.1 Uniform rectilinear motion This is rectilinear motion with constant velocity ($v$ = constant). Consequently, acceleration is zero ($a = 0$). The position equation is [27](#page=27): $$ x = x_0 + vt $$ [28](#page=28). #### 2.5.2 Uniformly accelerated rectilinear motion This is rectilinear motion with constant acceleration. The kinematic equations for motion along the x-axis with initial position $x_0$, initial velocity $v_0$, and constant acceleration $a$ are: * Velocity: $$ v = v_0 + at $$ [28](#page=28). * Position: $$ x = x_0 + v_0 t + \frac{1}{2} at^2 $$ [28](#page=28). * Velocity-position relation (eliminating time): $$ v^2 = v_0^2 + 2a(x - x_0) $$ [29](#page=29). > **Example:** A skier descends a straight slope with a constant acceleration of 2 m/s². If their speed at position 0 is 10 m/s, their speed after covering 100 m can be found using $v^2 = v_0^2 + 2a\Delta x$, resulting in $v \approx 22.36$ m/s. The time taken is then found from $v = v_0 + at$, yielding $t \approx 6.18$ s [29](#page=29). ### 2.6 Two-dimensional motion with constant acceleration When the acceleration is not in the same direction as the motion, a vector approach in two dimensions is necessary. The position, velocity, and acceleration vectors can be decomposed into x and y components [34](#page=34). For constant acceleration $\mathbf{a}$: * $a_x$ and $a_y$ are constant [35](#page=35). * The kinematic equations are applied independently to each component: * $v_x = v_{0x} + a_x t$ and $x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2$ [35](#page=35). * $v_y = v_{0y} + a_y t$ and $y = y_0 + v_{0y} t + \frac{1}{2} a_y t^2$ [35](#page=35). In vector form: * $\mathbf{v} = \mathbf{v}_0 + \mathbf{a} t$ [35](#page=35). * $\mathbf{r} = \mathbf{r}_0 + \mathbf{v}_0 t + \frac{1}{2} \mathbf{a} t^2$ [35](#page=35). #### 2.6.1 Projectile motion Projectile motion is a common two-dimensional motion with constant acceleration (due to gravity), assuming air resistance is negligible. It can be analyzed as the composition of a horizontal uniformly rectilinear motion and a vertical uniformly accelerated rectilinear motion [35](#page=35) [36](#page=36). * Horizontal motion: $a_x = 0$, $v_x = v_{0x} = v_0 \cos \theta$, $x = x_0 + (v_0 \cos \theta) t$ [36](#page=36). * Vertical motion: $a_y = -g$ (assuming positive y is upwards), $v_y = v_{0y} - gt = v_0 \sin \theta - gt$, $y = y_0 + (v_0 \sin \theta) t - \frac{1}{2} gt^2$ [36](#page=36). The trajectory of a projectile is a parabola [37](#page=37). Key parameters for projectile motion: * **Highest point:** Reached when $v_y = 0$. The time to reach the highest point is $t_{top} = \frac{v_0 \sin \theta}{g}$. The maximum height reached is $y_{max} = y_0 + \frac{(v_0 \sin \theta)^2}{2g}$ [37](#page=37). * **Range (R):** The horizontal distance covered before the projectile returns to its initial height. For symmetric parabolic motion ($y_{land} = y_0$), the total time of flight is $T = \frac{2 v_0 \sin \theta}{g}$, and the range is $R = \frac{v_0^2 \sin(2\theta)}{g}$. The maximum range occurs at a launch angle of $45^\circ$ for symmetric parabolas [39](#page=39) [40](#page=40). * **Asymmetric parabola:** If the landing height is different from the launch height, the trajectory is an asymmetric parabola, and the range and time of flight equations are modified [40](#page=40). > **Tip:** For maximum range in projectile motion, the optimal launch angle depends on the relative starting and landing heights. It's $45^\circ$ for symmetric trajectories, less than $45^\circ$ if starting higher than landing, and more than $45^\circ$ if starting lower than landing [44](#page=44). The three main factors influencing projectile motion are the initial angle, the initial velocity, and the initial height. Increasing the initial velocity generally has the most significant positive impact on range compared to increasing the angle or height [44](#page=44) [45](#page=45). > **Example:** A soccer ball is kicked with a speed of 80 m/s at an angle of 45° and lands 50 m away. Using the range formula $R = \frac{v_0^2 \sin(2\theta)}{g}$ (neglecting air resistance), we can verify this [40](#page=40). In real-world scenarios, air resistance can significantly affect projectile motion, causing the maximum range to occur at an angle less than $45^\circ$ and resulting in a steeper landing angle than the launch angle [46](#page=46). --- # Angular kinematics and kinetics This section explores the principles of rotational motion, defining angular displacement, velocity, and acceleration. ### 2.1 Basic concepts Rotation can be simplified by assuming that all mass elements of a body move in circular paths around a rotation axis, all with the same angular velocity. This approximation is generally valid for human body segments during most movements [51](#page=51). To describe the rotation of a point mass around a fixed axis, a reference line is established. This line is perpendicular to the rotation axis and rotates with the mass. The angle $\theta$ is formed by this reference line and the positive x-axis [51](#page=51). The relationship between arc length ($s$), radius ($r$), and angle ($\theta$) is given by: $s = r\theta$ [52](#page=52). where $\theta$ is measured in radians. Note that $\pi$ radians equals 180 degrees [52](#page=52). Angular displacement ($\Delta\theta$) is the change in angular position from $\theta_1$ at time $t_1$ to $\theta_2$ at time $t_2$: $\Delta\theta = \theta_2 - \theta_1$ [52](#page=52). A positive $\Delta\theta$ indicates counter-clockwise rotation, while a negative value indicates clockwise rotation [52](#page=52). Angular velocity ($\omega$) is defined as the rate of change of angular displacement over time. Average angular velocity ($\left\langle \omega \right\rangle$) is calculated as: $\left\langle \omega \right\rangle = \frac{\Delta\theta}{\Delta t} = \frac{\theta_2 - \theta_1}{t_2 - t_1}$ [52](#page=52). Instantaneous angular velocity ($\omega$) is the derivative of angular displacement with respect to time: $\omega = \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t} = \frac{d\theta}{dt}$ [52](#page=52). Both average and instantaneous angular velocities are vectors, with their direction determined by a specific rule (implied to be the right-hand rule) [52](#page=52). Average angular acceleration ($\left\langle \alpha \right\rangle$) is defined as: $\left\langle \alpha \right\rangle = \frac{\Delta\omega}{\Delta t}$ [52](#page=52). Instantaneous angular acceleration ($\alpha$) is the derivative of angular velocity with respect to time: $\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$ [52](#page=52). Angular acceleration is also a vector quantity [52](#page=52). ### 2.2 Determination of angles in biomechanics #### 2.2.1 The relative angle The relative angle measures the angle between the longitudinal axes of two body segments. It describes the orientation of one segment with respect to another, rather than its position in space [53](#page=53). To calculate the relative angle between two segments (e.g., thigh and shank to find the knee angle), the lengths of the sides of the triangle formed by the joint centers can be determined using the distance formula. For example, if the coordinates of the hip, knee, and ankle are known, the lengths $a$, $b$, and $c$ can be calculated. The cosine rule is then applied to find the included angle ($\theta$) [53](#page=53): $\cos \theta = \frac{b^2 + c^2 - a^2}{2bc}$ [54](#page=54). This angle can then be converted to degrees [54](#page=54). #### 2.2.2 The absolute angle The absolute angle indicates the orientation of a body segment with respect to a fixed reference (e.g., the horizontal). This is calculated by determining the angle of inclination of the segment's longitudinal axis relative to a horizontal line. Trigonometric functions like the tangent can be used, often requiring adjustments based on the quadrant of the angle [54](#page=54). ### 2.3 Uniform circular motion Uniform circular motion occurs when the angular velocity of a rotating object is constant, meaning the angular acceleration is zero. The velocity vector ($v$) maintains a constant magnitude but continuously changes direction, indicating acceleration is present [55](#page=55). Consider the path and hodograph of the velocity vector. By analyzing similar triangles in the path and hodograph, the relationship between changes in velocity and position can be established: $\frac{\Delta v}{v} = \frac{\Delta r}{r}$ [55](#page=55). Dividing by $\Delta t$ and taking the limit as $\Delta t \to 0$ yields: $\frac{dv}{dt} = \frac{v}{r} \frac{dr}{dt}$ [55](#page=55). which simplifies to: $a_n = v \omega = r \omega^2$ [55](#page=55). The direction of the centripetal acceleration ($a_n$) is always perpendicular to the velocity vector and directed towards the center of the circular path. This acceleration is essential for maintaining circular motion [55](#page=55). The linear velocity ($v$) along the circular path is given by: $v = \frac{ds}{dt} = r \frac{d\theta}{dt} = r\omega$ [56](#page=56). For a point at a radial distance $r$ from the axis of rotation undergoing circular motion with constant angular velocity $\omega$: $\omega(t) = \omega_0$ [56](#page=56). $\theta(t) = \theta_0 + \omega_0 t$ [56](#page=56). $v(t) = r\omega_0$ [56](#page=56). $a_n(t) = r\omega_0^2$ [56](#page=56). When angular velocity is constant, the time taken for one revolution (the period, $T$) is also constant: $T = \frac{2\pi}{\omega}$ [56](#page=56). Frequency ($f$) is the number of revolutions per unit time: $f = \frac{1}{T}$ [57](#page=57). The unit of frequency is Hertz (Hz), which is equivalent to 1/s [57](#page=57). ### 2.4 Uniformly accelerated rotational motion Uniformly accelerated rotational motion occurs when the angular velocity increases at a constant rate, meaning there is a constant angular acceleration. Analogous to linear uniformly accelerated motion, the kinematic equations are [57](#page=57): $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$ [57](#page=57). $\omega = \omega_0 + \alpha t$ [57](#page=57). $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ [57](#page=57). ### 2.5 Rotation of a rigid body about a fixed axis A rigid body is defined as a body where the distance between any pair of points remains constant. When a rigid body rotates in the xy-plane around a fixed z-axis, every point on the body traces a circular path in a plane perpendicular to the rotation axis [57](#page=57). Linear velocity ($v$) and acceleration ($a$) vectors are typically resolved into two perpendicular components: tangential (parallel to the circular path) and normal (radial, perpendicular to the path) [57](#page=57). The arc length ($s$) traversed by a point at radius $r$ at time $t$ is: $s = r\theta$ [58](#page=58). where $\theta$ is in radians [58](#page=58). The linear velocity vector ($v$), always tangent to the path, has a magnitude determined by the rate of change of position along the path: $v = \frac{ds}{dt} = r\omega$ [58](#page=58). The linear acceleration vector generally has two components: 1. **Tangential acceleration ($a_t$)**: Represents the rate of change of the magnitude of the velocity vector. $a_t = \frac{dv}{dt} = r \frac{d\omega}{dt} = r\alpha$ [58](#page=58). 2. **Normal acceleration ($a_n$)**: Represents the rate of change of the direction of the velocity vector. This is the centripetal acceleration, always directed towards the center of the circle. $a_n = \frac{v^2}{r} = r\omega^2$ [58](#page=58). For a body rotating at a constant speed ($v$), the tangential acceleration ($a_t$) is zero. The normal acceleration ($a_n$) is always present because the direction of the velocity vector is constantly changing [58](#page=58). The resultant acceleration vector ($a$) of a point on a body rotating about a fixed axis is the vector sum of the tangential and normal components: $\vec{a} = \vec{a}_t + \vec{a}_n$ [59](#page=59). The magnitude of the resultant acceleration is: $a = \sqrt{a_t^2 + a_n^2}$ [59](#page=59). where $\hat{t}$ and $\hat{n}$ are unit vectors in the tangential and normal directions, respectively [59](#page=59). ### 2.6 Relationship between linear and angular quantities Many human movements involve linear outcomes resulting from the angular motions of body parts. Understanding the relationship between linear and angular quantities is crucial [59](#page=59). From the arc length relation $s = r\theta$, for circular motion ($r$ = constant), we can derive: $\frac{ds}{dt} = r \frac{d\theta}{dt} \implies v = r\omega$ [59](#page=59). $\frac{dv}{dt} = r \frac{d\omega}{dt} \implies a_t = r\alpha$ [60](#page=60). $v^2 = (r\omega)^2 \implies a_n = \frac{v^2}{r} = r\omega^2$ [60](#page=60). The magnitude of the linear velocity of a point in a rotating body is equal to its distance from the rotation axis multiplied by the body's angular velocity. While all points on a body have the same angular velocity, their linear velocities may differ [59](#page=59). This relationship is vital in sports where objects are thrown or struck. For instance, a footballer can increase the ball's linear velocity by increasing the angular velocity of the lower leg segments or by increasing the length of the limb segments. In sports like golf, varying club lengths ($r$) and head angles allows for different ball distances [60](#page=60). The components of linear acceleration can be expressed as: $a_t = r\alpha$ [60](#page=60). $a_n = r\omega^2$ [60](#page=60). > **Example:** Technique of kicking a football > During a football kick, the leg moves in a complex sequence of rotations. The thigh rotates around the hip, followed by the lower leg extending around the knee. A high angular velocity of the lower leg results in a high linear velocity of the foot, crucial for ball speed. Experienced male footballers can achieve ball speeds of 20 to 30 meters per second, with foot speeds between 18 and 28 meters per second. Energy for the rapid angular acceleration of the lower leg is built up in earlier phases of the kick, with contributions from both the thigh and lower leg rotations [61](#page=61). --- # Linear and angular kinetics: forces and energy This section delves into the fundamental principles of linear and angular kinetics, exploring the relationships between forces, motion, work, and energy. ### 4.1 Newton's laws of motion Newton's laws of motion form the cornerstone of classical mechanics, explaining the relationship between an object's motion and the forces acting upon it [63](#page=63). #### 4.1.1 Newton's first law: the law of inertia Newton's first law states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. Mathematically, this can be expressed as [64](#page=64): $\sum_{i} \vec{F}_i = 0 \Leftrightarrow \vec{a} = 0$ [64](#page=64). #### 4.1.2 Newton's second law: the relationship between force and acceleration Newton's second law quantifies the relationship between force, mass, and acceleration. It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The SI unit of force is the newton (N), defined as $1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2$. The law is mathematically formulated as [64](#page=64) [65](#page=65): $$ \sum_{i} \vec{F}_i = m\vec{a} $$ [64](#page=64). This vector equation can be broken down into component equations for each axis: $$ \begin{cases} \sum_{i} F_{ix} = ma_x \\ \sum_{i} F_{iy} = ma_y \\ \sum_{i} F_{iz} = ma_z \end{cases} $$ [65](#page=65). #### 4.1.3 Newton's third law: action and reaction Newton's third law states that for every action, there is an equal and opposite reaction. When one object exerts a force on a second object, the second object exerts an equal and opposite force on the first. This can be expressed as [65](#page=65): $\vec{F}_{AB} = -\vec{F}_{BA}$ [65](#page=65). It is crucial to remember that action-reaction forces act on different bodies and thus do not cancel each other out [65](#page=65). Newton's laws of motion are valid in inertial reference frames, which are systems at rest or moving with constant velocity [65](#page=65). #### 4.1.4 Measuring forces Forces can be measured indirectly by observing their effects, such as deformations or changes in motion. A common method involves using a calibrated spring, based on Hooke's Law: $\vec{F} = k \Delta \vec{s}$ [66](#page=66). where $k$ is the spring constant and $\Delta s$ is the extension or compression of the spring [66](#page=66). ### 4.2 Key forces Several fundamental forces are frequently encountered in physics and biomechanics. #### 4.2.1 Gravitational force – weight Mass is an intrinsic property of an object, representing its inertia. In the Earth's gravitational field, an object also possesses weight, which is the force of gravity acting on its mass [66](#page=66). $\vec{G} = m\vec{g}$ [66](#page=66). where $\vec{g}$ is the acceleration due to gravity. The value of $g$ varies slightly with latitude and altitude. Weight is a vector force, while mass is a scalar measure of inertia [67](#page=67). #### 4.2.2 Normal force The normal force is a contact force exerted by a surface on an object, acting perpendicular to the surface. Its magnitude depends on the situation; it can be equal to, greater than, or less than the gravitational force [67](#page=67). * **Horizontal surface:** On a horizontal surface, the normal force typically counteracts the weight, unless other vertical forces are present. For example, if a force $\vec{F}$ is applied at an angle $\theta$ to the horizontal, the normal force $N$ is given by $N = W - F \sin \theta$ [68](#page=68). * **Inclined surface:** On an inclined plane, the normal force is equal to the component of the gravitational force perpendicular to the plane: $N = W \cos \theta$ [69](#page=69). #### 4.2.3 Tension force Tension is the pulling force transmitted axially by the means of a string, rope, cable, or similar one-dimensional object, or by each end of a rod. It acts along the direction of the string or rope [70](#page=70). #### 4.2.4 Friction force Friction is a force that opposes motion between surfaces in contact. It can be beneficial for activities like walking or detrimental by reducing efficiency [70](#page=70). * **Static friction ($f_s$):** This force prevents an object from starting to move. Its maximum value is given by $f_{s,\text{max}} = \mu_s N$, where $\mu_s$ is the static friction coefficient. The static friction force is variable, equaling the applied force up to its maximum value [71](#page=71). * **Kinetic friction ($f_k$):** This force opposes the motion of an object that is already moving. Its magnitude is given by $f_k = \mu_k N$, where $\mu_k$ is the kinetic friction coefficient. Typically, $\mu_s > \mu_k$ [72](#page=72). Friction coefficients depend on the materials in contact and their surface conditions [72](#page=72). #### 4.2.5 Ground reaction force In biomechanics, the ground reaction force is the force exerted by the ground on a body. It can be resolved into vertical (normal) and horizontal (shear) components. The anteroposterior component acts as friction enabling locomotion, while the vertical component is the normal force. The kinetic friction coefficient can be calculated from these components: $\mu_k = F_y / F_z$ [74](#page=74) [75](#page=75). #### 4.2.6 Joint reaction force Joint reaction forces are the net forces acting at a joint, resulting from the forces transmitted between body segments. They can be decomposed into compression and shear components. The direction of these forces is influenced by body posture [76](#page=76) [77](#page=77). #### 4.2.7 Muscle force Muscles generate force through contraction and act in a specific direction. Muscle forces can be resolved into stabilizing and rotating components relative to a joint's axis of rotation [77](#page=77) [78](#page=78). #### 4.2.8 Elastic forces Elastic forces arise from the deformation of elastic materials. For an ideal spring, Hooke's Law describes this relationship: $F = -kx$, where $k$ is the spring constant and $x$ is the displacement from equilibrium [79](#page=79). #### 4.2.9 Pseudoforces In non-inertial reference frames, fictitious forces like the centrifugal force are introduced to maintain the form of Newton's laws. However, this text advises avoiding pseudoforces to prevent confusion [80](#page=80) [81](#page=81). ### 4.3 Applications of linear kinetics #### 4.3.1 General problem-solving approach A systematic approach to solving problems in classical mechanics involves: 1. Defining a free-body diagram showing all external forces. 2. Choosing a suitable inertial reference frame. 3. Applying Newton's laws in vector form. 4. Resolving into component equations. 5. Solving the equations, paying attention to units. 6. Checking the plausibility of the result [81](#page=81). #### 4.3.2 Uniform circular motion example The centripetal force required for uniform circular motion is often provided by static friction, as seen in a car turning a corner [82](#page=82). #### 4.3.3 Dynamics of an inclined plane example Analyzing the motion of a skier down an inclined slope involves resolving gravitational forces and accounting for kinetic friction (#page=83, 84) [83](#page=83) [84](#page=84). ### 4.4 Work and energy Work and energy are concepts that describe the capacity to do work. #### 4.4.1 The concept of work Work is done when a force causes an object to move. If the force $\vec{F}$ acts over a displacement $\vec{d}$, the work $W$ is defined as the scalar product of the force and displacement: $W = \vec{F} \cdot \vec{d}$ [86](#page=86). If the force and displacement are in the same direction, $W = Fd$. If they are at an angle $\theta$, $W = Fd \cos \theta$ [86](#page=86). Positive work is done when the force component is in the direction of displacement ($0^\circ \le \theta < 90^\circ$). Zero work is done when the force is perpendicular to the displacement ($\theta = 90^\circ$). Negative work is done when the force opposes the displacement ($90^\circ < \theta \le 180^\circ$). The SI unit of work is the joule (J) [87](#page=87). #### 4.4.2 Work done by a variable force For a variable force $F(x)$ acting in one dimension, the work done over a displacement from $x_1$ to $x_2$ is given by the integral: $$ W = \int_{x_1}^{x_2} F(x) \, dx $$ (#page=88, 89) [88](#page=88) [89](#page=89). Graphically, this is the area under the force-displacement curve. For a general force $\vec{F}(\vec{r})$ in three dimensions, the work done is a line integral [89](#page=89): $$ W = \int_{a}^{b} \vec{F} \cdot d\vec{r} $$ [91](#page=91). #### 4.4.3 Work-energy theorem The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy: $W = \Delta K$ (#page=92, 93) [92](#page=92) [93](#page=93). where kinetic energy $K$ is defined as: $$ K = \frac{1}{2}mv^2 $$ [92](#page=92). #### 4.4.4 Work rate or power Power is the rate at which work is done. Average power is $P = W/t$, and instantaneous power is $P = dW/dt$. The SI unit of power is the watt (W), where $1 \, \text{W} = 1 \, \text{J/s}$ [95](#page=95). #### 4.4.5 Potential energy of a system * **Conservative and non-conservative forces:** A force is conservative if the work it does on an object moving between two points is independent of the path taken and if the net work done over a closed loop is zero (e.g., gravity, elastic force). Forces for which this is not true are non-conservative (e.g., friction) [97](#page=97). * **Potential energy:** For conservative forces, potential energy ($U$) is associated with the configuration or position of a system. The change in potential energy is the negative of the work done by the conservative force: $\Delta U = -W$ (#page=98, 99). For example, the gravitational potential energy near the Earth's surface is $U = mgh$ [98](#page=98) [99](#page=99). The relationship between force and potential energy is given by: $F(x) = -\frac{dU}{dx}$ [99](#page=99). or in vector form: $\vec{F} = -\nabla U$ [99](#page=99). #### 4.4.6 Law of conservation of mechanical energy For systems where only conservative forces do work, the total mechanical energy ($E = K + U$) remains constant (#page=102, 103). This means kinetic energy can be converted into potential energy and vice versa, but their sum is conserved . $E = K + U = \text{constant}$ . * **Ideal spring:** In a system with an ideal spring and mass on a frictionless surface, $E = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \text{constant}$ (#page=103, 104) . * **Pendulum:** For small oscillations, a pendulum also exhibits conservation of mechanical energy. #### 4.4.7 Law of conservation of total energy When non-conservative forces (like friction) are present, mechanical energy is not conserved but is converted into other forms of energy (e.g., heat, internal energy). The principle of conservation of total energy states that energy can neither be created nor destroyed, only transformed from one form to another. Einstein's mass-energy equivalence, $E = mc^2$, shows that mass is also a form of energy . #### 4.4.8 Energy fluctuations of the center of mass during locomotion During walking, kinetic and gravitational potential energies fluctuate out of phase, with kinetic energy converting to potential energy as the body rises and vice versa. During running, these energies fluctuate in phase . ### 4.5 Linear momentum and impulse Linear momentum (or quantity of motion) is a vector quantity defined as the product of an object's mass and its velocity: $\vec{p} = m\vec{v}$ . The rate of change of linear momentum is equal to the net force acting on the object (Newton's second law in terms of momentum): $$ \vec{F} = \frac{d\vec{p}}{dt} $$ . For a system of particles, the total linear momentum is the sum of individual momenta, which is equal to the total mass times the velocity of the center of mass: $\vec{P} = M\vec{v}_{cm}$. The rate of change of the total linear momentum is equal to the sum of external forces : $$ \sum_{i} \vec{F}_{\text{ext},i} = \frac{d\vec{P}}{dt} $$ . #### 4.5.1 Law of conservation of linear momentum If the net external force acting on a system is zero, its total linear momentum remains constant. This is a fundamental principle in physics, particularly useful for analyzing collisions . #### 4.5.2 Collisions: impulse of a force A collision involves a sudden, large force acting over a short time, known as an impulsive force. The impulse ($J$) of a force is the integral of the force over the time interval during which it acts : $$ \vec{J} = \int_{t_i}^{t_f} \vec{F} \, dt $$ . The impulse-momentum theorem states that the impulse applied to an object is equal to the change in its linear momentum: $\vec{J} = \Delta \vec{p}$ . #### 4.5.3 Conservation of momentum in collisions During collisions, the total linear momentum of the system is conserved, provided no external impulsive forces act (#page=120, 121). Collisions are classified based on kinetic energy conservation : * **Elastic collisions:** Kinetic energy is conserved. $\frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2$ . * **Inelastic collisions:** Kinetic energy is not conserved. * **Perfectly inelastic collisions:** The colliding bodies stick together after impact, moving as a single unit (#page=121, 124) . #### 4.5.4 One-dimensional collision of two bodies For a head-on collision between two bodies, both momentum and (if elastic) kinetic energy are conserved, allowing for the calculation of final velocities (#page=122, 123) . #### 4.5.5 Two- and three-dimensional collisions In multi-dimensional collisions, momentum is conserved independently in each direction. For elastic collisions, kinetic energy is also conserved . ### 4.6 Angular kinetics Angular kinetics deals with the causes of rotational motion. #### 4.6.1 Torque (moment of force) Torque ($\vec{\tau}$) is the rotational equivalent of force. It is defined as the cross product of the position vector ($\vec{r}$) from the pivot point to the point of force application and the force vector ($\vec{F}$): $$ \vec{\tau} = \vec{r} \times \vec{F} $$ . The magnitude of torque is $\tau = rF \sin \theta$, where $\theta$ is the angle between $\vec{r}$ and $\vec{F}$. The term $r_{\perp} = r \sin \theta$ is the lever arm or moment arm . #### 4.6.2 Relationship between torque and rotational inertia The rotational equivalent of Newton's second law relates net torque to angular acceleration ($\alpha$) and the moment of inertia ($I$): $$ \sum \vec{\tau}_{\text{ext}} = I \vec{\alpha} $$ . The moment of inertia ($I$) is the rotational analog of mass, representing resistance to rotational acceleration. It depends on the mass distribution relative to the axis of rotation . #### 4.6.3 Moment of inertia The moment of inertia of a continuous mass distribution about an axis is given by: $$ I = \int R^2 \, dm $$ . For a system of discrete point masses: $$ I = \sum_{i=1}^{n} m_i R_i^2 $$ . #### 4.6.4 Parallel-axis theorem (Steiner's theorem) Steiner's theorem relates the moment of inertia about an axis through the center of mass ($I_{CM}$) to the moment of inertia about a parallel axis ($I_P$) at a distance $d$: $I_P = I_{CM} + Md^2$ . #### 4.6.5 Rotational kinetic energy The kinetic energy of a rotating body is given by: $$ K_{\text{rot}} = \frac{1}{2}I\omega^2 $$ . where $\omega$ is the angular velocity. For a rolling object (without slipping), the total kinetic energy is the sum of translational and rotational kinetic energies: $K = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}\omega^2$ . #### 4.6.6 Angular momentum of a point mass Angular momentum ($\vec{l}$) of a point mass about an origin O is defined as: $$ \vec{l} = \vec{r} \times \vec{p} = \vec{r} \times (m\vec{v}) $$ . The relationship between angular momentum and torque is: $$ \vec{\tau} = \frac{d\vec{l}}{dt} $$ . #### 4.6.7 Angular momentum of a system of particles For a system of particles, the total angular momentum $\vec{L}$ about a fixed point is the sum of individual angular momenta. The rate of change of total angular momentum is equal to the net external torque: $$ \sum \vec{\tau}_{\text{ext}} = \frac{d\vec{L}}{dt} $$ . #### 4.6.8 Angular momentum of a rigid body For a rigid body rotating about a fixed axis, the component of angular momentum along that axis is $L_{\text{axis}} = I_{\text{axis}}\omega$ . #### 4.6.9 Law of conservation of angular momentum If the net external torque on a system is zero, its total angular momentum remains constant. This is analogous to the conservation of linear momentum . $L = I\omega = \text{constant}$ . --- # Statics of the human body This section explores the principles of statics as they apply to the human body, focusing on equilibrium conditions, the center of gravity, types of equilibrium, the base of support, and biomechanical applications like pulley systems and force analysis within body segments and joints . ### 5.1 Equilibrium conditions For a body to be in static equilibrium, the net external force acting on it must be zero, and the net external torque about any point must also be zero. This means that the vector sum of all external forces and the vector sum of all external torques must be zero. In three-dimensional motion, this translates to six scalar equilibrium conditions . ### 5.2 Determining the center of gravity The gravitational force, or weight, is crucial in biomechanics as it influences the development and function of the musculoskeletal system. The entire gravitational force acting on an object can be considered to act at its center of gravity (CG). The center of mass (CM) is defined as the point where all the mass of an object is concentrated. For biomechanical applications, the CG and CM are often considered the same point . #### 5.2.1 Determining CG for homogeneous and non-homogeneous bodies For homogeneous bodies, the CG can be determined by symmetry. For non-homogeneous or irregular bodies, the body is divided into segments, and the CG of each segment is determined and then combined to find the CG of the whole body . * **Rule for combining CGs:** The CG of a system of two masses ($m_1$ and $m_2$) with CGs ($Z_1$ and $Z_2$) divides the line segment $Z_1Z_2$ such that the distances from the combined CG ($Z$) to $Z_1$ and $Z_2$ are inversely proportional to the masses . $$ \frac{ZZ_1}{ZZ_2} = \frac{m_2}{m_1} $$ . * **General formulas for CG coordinates:** If the masses and coordinate systems of the segments are known, the coordinates of the total body's CG can be calculated using integrals: $$ x_{CG} = \frac{\int x \, dm}{\int dm}, \quad y_{CG} = \frac{\int y \, dm}{\int dm}, \quad z_{CG} = \frac{\int z \, dm}{\int dm} $$ . For a body composed of a finite number of point masses ($m_i$): $$ x_{CG} = \frac{\sum_{i=1}^{n} m_i x_i}{\sum_{i=1}^{n} m_i}, \quad y_{CG} = \frac{\sum_{i=1}^{n} m_i y_i}{\sum_{i=1}^{n} m_i}, \quad z_{CG} = \frac{\sum_{i=1}^{n} m_i z_i}{\sum_{i=1}^{n} m_i} $$ . It is important to note that the CG of a body does not necessarily lie within its physical boundaries . #### 5.2.2 The center of gravity of the human body Unlike rigid bodies, the human body does not have a fixed CG. Its location varies between individuals based on their build and can change for a specific person depending on the relative positions of their limbs during physical activity. Any movement of body parts results in a change in the body's overall CG. In the anatomical position (standing upright with arms at the sides), the human body's CG is located in the pelvis, anterior to the second sacral vertebra . #### 5.2.3 Experimental determination of the body's center of gravity (in vivo) The CG can be experimentally determined using scales. This involves measuring normal forces and applying the torque equilibrium condition . * **Example:** A person lies on a plank supported at one end and a scale at the other. The scale reading ($m_{afl}$) combined with the total length ($L$), person's mass ($m$), plank's mass ($M$), and the position of the scale's support allows for the calculation of the CG's position ($x$) relative to the head support . $$ x = \frac{(L \cdot m_{afl} \cdot g) - (L \cdot M \cdot g)}{m \cdot g} = L \frac{m_{afl} - M}{m} $$ . #### 5.2.4 Experimental determination of the partial center of gravity of the arms The position of the CG can be determined for specific body segments by repeating the scale measurement with different limb positions. By comparing the scale readings when arms are along the body versus when they are moved, the displacement ($d$) of the arms' CG can be calculated . * **Example:** The distance of the arms' CG from the shoulder joint is half of the displacement ($d/2$) . #### 5.2.5 Examples of center of gravity determination * **CG of the arm:** To determine the CG of the arm, the masses and positions of its components (hand, forearm, upper arm) are used to calculate the combined CG of the arm, typically measured from the shoulder . $$ x_{MC} = \frac{\sum m_i x_i}{\sum m_i} $$ . * **CG of a bent leg:** The CG of a bent leg can be determined by knowing the coordinates and relative weights of the thigh, lower leg, and foot, measured from the ground or hip joint . $$ x_{CG} = \frac{\sum W_i x_i}{\sum W_i}, \quad y_{CG} = \frac{\sum W_i y_i}{\sum W_i} $$ . ### 5.3 Forms of equilibrium When a body in static equilibrium is slightly disturbed, there are three possibilities: * **Stable equilibrium:** The body returns to its original position. This occurs when the CG rises upon disturbance, increasing potential energy . * **Labile (unstable) equilibrium:** The body moves further away from its original position. This occurs when the CG lowers upon disturbance, decreasing potential energy . * **Neutral equilibrium:** The body assumes a new equilibrium position. Every position is an equilibrium state . A **metastable equilibrium** occurs when a body behaves stably under small disturbances but becomes unstable beyond a certain threshold. Stability is generally increased by a lower CG . #### 5.3.1 The base of support and stability The base of support is the area on the ground defined by the points of contact with the supporting surface. A body is in equilibrium if the vertical projection of its CG falls within the base of support . * **Stability criteria:** * **Tilting angle ($\alpha$):** Defined by $\tan \alpha = \frac{h_{CG}}{d_{CG}}$, where $h_{CG}$ is the height of the CG and $d_{CG}$ is the horizontal distance from the CG to the edge of the base of support. A larger tilting angle indicates greater stability . * **Moment of the stabilizing couple ($mgu$):** This is the moment created by the gravitational force and the normal force that opposes tilting. A larger moment indicates greater stability . These two criteria are complementary, and the choice of which to apply depends on the specific situation . ### 5.4 External forces exerted with pulleys and hoists #### 5.4.1 Pulleys A pulley is a wheel with a groove for a rope, rotatable around an axle . * **Fixed pulley:** The support of the pulley is fixed . * **Movable pulley:** The pulley hangs loosely and moves with the load. For a movable pulley supporting a load, the force in the rope ($F_{rope}$) is related to the load's weight ($G_{last}$) by : $$ 2F_{rope} = G_{last} $$ . #### 5.4.2 Hoists (Tackles) A hoist is a combination of fixed and movable pulleys designed to gain mechanical advantage. The external force ($F$) required to lift a load with mass ($m_{last}$) using a hoist with $n$ movable pulleys and a total mass of movable pulleys ($m_{pulley}$) is : $$ F = \frac{(m_{last} + m_{pulley})g}{2^n} $$ . * **Example:** A hoist with one movable and two fixed pulleys supporting a load of mass $m_{last}$ and movable pulleys of mass $m_{pulley}$ requires an external force of: $$ F = \frac{(m_{last} + m_{pulley})g}{2^1} $$ . #### 5.4.3 External forces on a body part using pulleys and hoists Pulley systems can be used to apply traction to body parts. The resultant of external forces on a body part must be zero for equilibrium, with internal forces (muscles, joints) counteracting them . * **Traction of the cervical vertebrae:** A hoist system with one movable and two fixed pulleys can be used for cervical traction. The traction force on the vertebrae is the sum of the rope tensions and the weight of the head and frame, balanced by the reaction force from the vertebrae . $$ F_{vertebrae} = F_{head+frame+pulley} + 3F_{rope} $$ . If the rope tension is generated by a weight $m$, then $F_{rope} = mg$. $$ F_{vertebrae} = G_{head+frame+pulley} + 3mg $$ . ### 5.5 Determining internal forces: The biomechanical system The equilibrium of a body segment is a result of both external forces and internal forces such as muscle forces and joint reaction forces. To analyze muscle forces, body segments are treated as free bodies, making muscle forces external forces on that segment. Muscle forces are typically larger than external forces due to shorter lever arms . * **Muscle structure:** Striated muscles consist of muscle fibers, myofibrils, and sarcomeres, which contain actin and myosin filaments. Muscle contraction occurs through the shortening of sarcomeres as actin filaments slide between myosin filaments . #### 5.5.1 Determining muscle force and reaction force using parallel force systems In a parallel force system, moments are calculated about the pivot point, and equilibrium conditions are applied to the force components and moments along the axis of the forces . #### 5.5.2 Resolving forces into horizontal and vertical components Forces are typically resolved into horizontal (x) and vertical (y) components. The equilibrium of moments about the pivot point is used to determine muscle force, while the force equilibrium equations yield the x and y components of the reaction force at the pivot . #### 5.5.3 Resolving forces along axes belonging to a body part A coordinate system aligned with the body part's length axis is used. Muscle force is resolved into stabilizing (along the length axis) and rotating components. The stabilizing component exerts pressure on the joint and contributes to joint stability, while the rotating component creates or counteracts rotation . #### 5.5.4 Assumptions and limitations in analyzing human joints * Anatomical axes of rotation for joints are known . * Muscle attachment points are known . * The line of action of muscle tension is known . * Weights and CGs of body segments are known . * Friction in joints is neglected . * Only two-dimensional problems are studied . ### 5.6 Applications of statics in biomechanics #### 5.6.1 Influence of the patella on quadriceps function The patella acts as a pulley, changing the direction of the quadriceps force and increasing the moment arm for knee extension. This results in a larger rotating component of the muscle force and a smaller stabilizing (compressive) component on the tibiofemoral joint. Without a patella, the quadriceps would need to exert a significantly larger force to produce the same extension moment . * **Example:** Calculating quadriceps force and its stabilizing component with and without a patella. The presence of the patella significantly reduces the required muscle force and joint compression . #### 5.6.2 Example of a parallel force system analysis This involves calculating the biceps muscle force and elbow joint reaction force required to maintain the forearm in a horizontal position . $$ \sum F_y = F_{sp} - F_r - W = 0 $$ . $$ \sum \tau_O = a F_{sp} - b W = 0 $$ . * **Result:** The biceps muscle exerts a force of 75 N, and the elbow joint reaction force is 55 N downwards. The angle of the forearm relative to the vertical influences the translational and rotational components of the muscle force, affecting joint stability . #### 5.6.3 Example of resolving forces into horizontal and vertical components This involves determining the hip joint reaction force for a standing person . * **Standing on both legs:** The weight of the upper body is distributed equally between the hip joints . $$ 2F_{hip} = W_{upper\_body} $$ . The reaction force in each hip is approximately 306.15 N . * **Standing on one leg:** A free-body diagram of the leg is used. The hip abductor muscles exert a force ($F_{sp}$), and the ground provides a normal reaction force ($N$) . $$ \sum F_y = N \sin(71^\circ) + F_{sp} \sin(\theta) - W_b - W_{body} = 0 $$ . $$ \sum F_x = N \cos(71^\circ) + F_{sp} \cos(\theta) - W_b = 0 $$ . $$ \sum \tau_O =... $$ . The resulting hip joint force can be very large, over 2000 N, especially during single-leg stance . #### 5.6.4 Example of resolving forces along axes belonging to a body part This analyzes the forces required to maintain a flexed posture, such as bending forward at the waist . * **Forward bending:** The erector spinae muscles exert a force ($F_{sp}$) to counteract the gravitational force on the torso ($W$). The reaction force in the hip joint is then calculated . $$ \sum \tau_{hip} =... $$ . The muscle force and hip joint reaction force increase significantly with added loads. Correct lifting techniques, involving keeping the weight close to the body and maintaining a straight or slightly curved back, are crucial for minimizing stress on the lower back . --- ## Common mistakes to avoid - Review all topics thoroughly before exams - Pay attention to formulas and key definitions - Practice with examples provided in each section - Don't memorize without understanding the underlying concepts

| Term | Definition | |------|------------| | Mechanics | The branch of physics concerned with the motion of bodies and the forces that cause such motion. | | Biomechanics | The application of mechanical principles to biological systems, such as the study of human and animal movement. | | Statics | The branch of mechanics that studies bodies at rest or in equilibrium under the action of forces. | | Dynamics | The branch of mechanics that studies the motion of bodies and the forces that cause it. | | Kinematics | The study of the motion of bodies without considering the forces that cause the motion; it focuses on the geometric properties of motion. | | Kinetics | The study of the forces responsible for causing motion. | | Linear Motion | A type of motion where all parts of a body move the same distance in the same time and direction. Also known as translation. | | Angular Motion | Motion that occurs when all parts of a body move through the same angle in the same time interval, rotating around an axis. Also known as rotation. | | General Motion | Motion that occurs when translation and rotation happen simultaneously. | | Vector | A quantity having both magnitude and direction, represented graphically by an arrow. | | Scalar | A quantity that has magnitude only, such as length, mass, or temperature. | | Velocity | The rate at which an object changes its position; it is a vector quantity with both magnitude and direction. | | Acceleration | The rate at which the velocity of an object changes over time; it is a vector quantity. | | Free Fall | The motion of a body falling freely under the influence of gravity alone, neglecting air resistance. | | Projectile Motion | The motion of an object thrown or projected into the air, subject to gravity and air resistance. | | Work | The energy transferred when a force moves an object over a distance; defined as the dot product of force and displacement. | | Kinetic Energy | The energy an object possesses due to its motion, calculated as $1/2 \ast m \ast v^2$. | | Potential Energy | Stored energy that an object possesses due to its position or state, such as gravitational potential energy ($m \ast g \ast h$) or elastic potential energy ($1/2 \ast k \ast x^2$). | | Power | The rate at which work is done or energy is transferred. | | Conservation of Mechanical Energy | The principle that in a conservative system, the total mechanical energy (kinetic plus potential) remains constant. | | Linear Momentum | The product of an object's mass and its velocity; it is a vector quantity ($p = m \ast v$). | | Impulse | The change in momentum of an object, equal to the integral of force over time or the product of average force and the time interval. | | Conservation of Linear Momentum | The principle that the total momentum of a system remains constant if no external forces act on it. | | Angular Momentum | The rotational equivalent of linear momentum, defined as the cross product of the position vector and linear momentum ($l = r \times p$). | | Torque (Moment of Force) | The rotational equivalent of force, defined as the cross product of the position vector and the force vector ($ \tau = r \times F $). | | Moment of Inertia (Rotational Inertia) | The rotational equivalent of mass, representing an object's resistance to changes in its rotational motion. It depends on mass and its distribution relative to the axis of rotation. | | Conservation of Angular Momentum | The principle that the total angular momentum of a system remains constant if no external torques act on it. | | Statics | The study of forces and their effects on bodies at rest or in equilibrium. | | Equilibrium | A state where the net force and net torque acting on an object are zero, resulting in no acceleration or angular acceleration. | | Center of Gravity | The point where the entire weight of an object can be considered to act. | | Base of Support | The area defined by the points of contact of an object with the ground. | | Free Body Diagram | A diagram showing an object and all the external forces acting upon it. | | Normal Force | The force exerted by a surface perpendicular to the surface on an object in contact with it. | | Friction Force | A force that opposes motion or the tendency of motion between surfaces in contact. | | Centripetal Force | A force that acts on a body moving in a circular path and is directed towards the center around which the body is moving. | | Centrifugal Force | A fictitious outward force experienced by an object in a rotating frame of reference. | | Impulse-Moment Theory | A theory that relates the impulse of a force to the change in an object's momentum. | | Hodograph | A curve traced by the tip of a vector representing velocity or acceleration over time. | | Rotational Inertia | The resistance of an object to changes in its rotational motion, analogous to mass in linear motion. | | Parallel Axis Theorem (Steiner's Theorem) | A theorem that relates the moment of inertia of a body about an axis to the moment of inertia about a parallel axis passing through the center of mass. | | Kinematics of Rotation | The study of the geometric aspects of rotational motion, including angular displacement, velocity, and acceleration. | | Kinetics of Rotation | The study of the rotational motion of bodies and the forces and torques that cause these motions. | | Torque | The rotational equivalent of force, causing or tending to cause rotation. | | Angular Acceleration | The rate of change of angular velocity. | | Angular Velocity | The rate of change of angular displacement. | | Conservation of Total Energy | The principle that the total energy of an isolated system remains constant; energy can be transformed from one form to another but cannot be created or destroyed. | | Shear Force | A force acting parallel to a surface, tending to cause one part of the body to slide relative to another. | | Reaction Force | A force exerted by a surface or body in response to an applied force, according to Newton's third law. | | Muscle Force | The force generated by the contraction of muscles, which is crucial for movement. | | Joint Reaction Force | The force exerted across a joint due to the combined actions of muscles, ligaments, and external loads. | | Work-Energy Theorem | The theorem stating that the net work done on an object is equal to the change in its kinetic energy. | | Rotational Kinetic Energy | The energy of a body due to its rotation, calculated as $1/2 \ast I \ast \omega^2$. | | Lever Arm | The perpendicular distance from the axis of rotation to the line of action of a force. | | Stability | The ability of an object to maintain its equilibrium position when subjected to external disturbances. | | Metastable Equilibrium | A state of equilibrium that is stable for small disturbances but becomes unstable beyond a certain threshold. |