Fisica appunti.docx

# Introduction to units and dimensional analysis A number devoid of its unit is meaningless, emphasizing the critical importance of units in physics and the use of dimensional analysis to verify the correctness of equations. ### 1.1 The significance of units * In physics, a numerical value is incomplete and lacks context without its associated unit. For example, a speed of 5 has no meaning until it is specified as 5 meters per second ($5 \, \text{m/s}$) or 5 kilometers per hour ($5 \, \text{km/h}$). ### 1.2 Dimensional analysis Dimensional analysis is a powerful technique used to verify the physical correctness of equations. It operates on the principle that any physically valid equation must have consistent dimensions on both sides. #### 1.2.1 Fundamental dimensions The primary physical quantities in mechanics are typically expressed in terms of three fundamental dimensions: * Length: denoted by $[L]$ * Mass: denoted by $[M]$ * Time: denoted by $[T]$ #### 1.2.2 Dimensional equations To perform dimensional analysis, we represent physical quantities by their dimensions. For instance: * Velocity has dimensions of length per time: $[v] = [L]/[T]$. * Acceleration has dimensions of length per time squared: $[a] = [L]/[T]^2$. * Force has dimensions of mass times acceleration: $[F] = [M][L]/[T]^2$. #### 1.2.3 Verifying equations An equation is dimensionally correct if the dimensions of all terms on one side are identical to the dimensions of all terms on the other side. > **Tip:** Dimensional analysis can help identify errors in calculations or the form of an equation, but it cannot determine the exact numerical constants (like dimensionless coefficients) within an equation. **Example:** Consider the equation for average velocity: $v_{avg} = \frac{x_2 - x_1}{t_2 - t_1}$. The dimensions of the numerator are $[L] - [L] = [L]$. The dimensions of the denominator are $[T] - [T] = [T]$. Therefore, the dimensions of the average velocity are $\frac{[L]}{[T]}$, which matches the expected dimensions of velocity. #### 1.2.4 Application in physics * **Free fall:** In free fall under gravity, the acceleration due to gravity $g$ is independent of mass. This can be inferred from dimensional analysis. If an equation for a quantity dependent on mass were dimensionally correct, the mass term would need to cancel out, suggesting its irrelevance to that particular quantity. * **Proportionality:** Dimensional analysis is useful for determining the proportionality of quantities. For example, if the speed $v$ of an object is related to a radius $R$ and a frequency $f$ by $v = c R^a f^b$, dimensional analysis can help find the exponents $a$ and $b$ by equating the dimensions on both sides. If $[v] = [L]/[T]$, $[R] = [L]$, and $[f] = 1/[T]$, then $[L]/[T] = [c] [L]^a (1/[T])^b$. For this to hold true, $a$ must be 1, and $b$ must be 1. This suggests that the speed is proportional to the radius and the frequency, so $v \propto R f$. ### 1.3 Measurement and uncertainty All measurements in physics inherently involve some degree of uncertainty, which depends on the circumstances and the precision of the measuring instruments. This uncertainty must be considered when reporting results. --- # Motion in one, two, and three dimensions Motion in one, two, and three dimensions provides the foundational principles for describing how objects move through space and time. ## 2 Motion in one, two, and three dimensions ### 2.1 Kinematics: Describing Motion Kinematics is the study of motion without considering its causes. #### 2.1.1 Position, Velocity, and Acceleration * **Position:** To describe motion, we need to define an object's position. In one dimension, this is typically represented by a single coordinate. In higher dimensions, it requires multiple coordinates. * **Average Velocity:** The average velocity is the displacement divided by the time interval over which the displacement occurred. $$ v_{\text{avg}} = \frac{\Delta x}{\Delta t} = \frac{x_2 - x_1}{t_2 - t_1} $$ * **Instantaneous Velocity:** The instantaneous velocity 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, which is the derivative of position with respect to time. $$ v(t) = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt} $$ * **Average Acceleration:** The average acceleration is the change in velocity divided by the time interval over which the velocity changed. $$ a_{\text{avg}} = \frac{\Delta v}{\Delta t} = \frac{v_2 - v_1}{t_2 - t_1} $$ * **Instantaneous Acceleration:** The instantaneous acceleration is the acceleration of an object at a specific moment in time. It is the derivative of velocity with respect to time, or the second derivative of position with respect to time. $$ a(t) = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt} = \frac{d^2x}{dt^2} $$ #### 2.1.2 Free Fall Free fall is the motion of an object under the sole influence of gravity. In free fall, the acceleration due to gravity ($g$) is constant and directed downwards. The mass of the object does not affect its acceleration due to gravity. * The equation for velocity in free fall, assuming initial velocity $v_0 = 0$ and starting at time $t=0$, is: $$ v(t) = -gt $$ * The equation for position in free fall, assuming initial position $y(0) = h$ and initial velocity $v_0 = 0$, is: $$ y(t) = -\frac{1}{2}gt^2 + h $$ * The time it takes for an object to fall from a height $h$ to the ground (where $y(t) = 0$) is: $$ t = \sqrt{\frac{2h}{g}} $$ #### 2.1.3 Motion with Constant Acceleration For motion with constant acceleration, kinematic equations can be derived relating displacement, initial velocity, final velocity, acceleration, and time. If $a$ is the constant acceleration: * $$ v = v_0 + at $$ * $$ x = x_0 + v_0t + \frac{1}{2}at^2 $$ * $$ v^2 = v_0^2 + 2a(x - x_0) $$ ### 2.2 Vectors in Motion Vectors are essential for describing motion in two and three dimensions because they possess both magnitude and direction. #### 2.2.1 Vector Representation * **Position Vector:** A vector pointing from the origin to the object's position. * **Velocity Vector:** A vector representing the instantaneous velocity, tangent to the path of motion. * **Acceleration Vector:** A vector representing the instantaneous acceleration. #### 2.2.2 Vector Operations * **Addition:** Vectors can be added graphically (tip-to-tail method) or using components. Only vectors of the same dimension can be added. $$ \vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k} $$ * **Subtraction:** Vector subtraction is equivalent to adding the negative of the second vector. $$ \vec{A} - \vec{B} = \vec{A} + (-\vec{B}) $$ * **Scalar Multiplication:** Multiplying a vector by a scalar changes its magnitude but not its direction (unless the scalar is negative). * **Scalar (Dot) Product:** The scalar product of two vectors results in a scalar quantity. It is used to find the projection of one vector onto another. $$ \vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z = |\vec{A}| |\vec{B}| \cos \theta $$ The scalar product is zero if the vectors are orthogonal. * **Vector (Cross) Product:** The vector product of two vectors results in a vector that is perpendicular to both original vectors. Its direction is given by the right-hand rule. $$ \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} $$ #### 2.2.3 Motion in 3D In three dimensions, position, velocity, and acceleration are all represented by vectors with $x$, $y$, and $z$ components. These components change with time independently, according to the principles of kinematics. * Position: $ \vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k} $ * Velocity: $ \vec{v}(t) = \frac{d\vec{r}}{dt} = \frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j} + \frac{dz}{dt}\hat{k} $ * Acceleration: $ \vec{a}(t) = \frac{d\vec{v}}{dt} = \frac{d^2x}{dt^2}\hat{i} + \frac{d^2y}{dt^2}\hat{j} + \frac{d^2z}{dt^2}\hat{k} $ ### 2.3 Projectile Motion in 2D Projectile motion is a specific case of 2D motion where an object is launched with an initial velocity and then moves under the influence of gravity alone. * The motion can be analyzed by separating it into independent horizontal (x) and vertical (y) components. * **Horizontal Motion:** Assuming no air resistance, the horizontal velocity is constant ($v_x = v_0 \cos \alpha$). * $ x(t) = (v_0 \cos \alpha)t $ * **Vertical Motion:** The vertical motion is governed by gravity, similar to free fall ($a_y = -g$). * $ v_y(t) = v_0 \sin \alpha - gt $ * $ y(t) = (v_0 \sin \alpha)t - \frac{1}{2}gt^2 $ * **Time of Flight:** The total time the projectile spends in the air can be found by setting the vertical displacement to zero ($y(t) = 0$) and solving for $t$. * **Range:** The horizontal distance traveled by the projectile. * **Maximum Height:** The highest vertical position reached by the projectile. > **Tip:** When analyzing projectile motion, always assume the x-component of acceleration is zero (neglecting air resistance) and the y-component is $-g$. > **Example:** A ball is kicked with an initial velocity of 20 meters per second at an angle of 30 degrees above the horizontal. To find its range and maximum height, you would use the projectile motion equations, treating the horizontal and vertical components of motion separately. ### 2.4 Relative Motion Relative motion considers how motion appears from different reference frames. * **Inertial Reference Systems:** These are reference frames in which Newton's laws of motion hold true. An inertial system is one that is at rest or moving with constant velocity relative to another inertial system. * **Relative Velocity:** If frame S' moves with velocity $\vec{v}_{S'}$ relative to frame S, and an object has velocity $\vec{v}_{obj/S'}$ in frame S', then its velocity in frame S is $\vec{v}_{obj/S} = \vec{v}_{obj/S'} + \vec{v}_{S'}$. * **Galilean Relativity:** The laws of physics are the same in all inertial reference systems. > **Tip:** When dealing with relative motion, carefully identify your reference frames and the velocities associated with each. > **Example:** Imagine a person walking on a moving train. Their velocity relative to the train is different from their velocity relative to the ground. ### 2.5 Non-Inertial Reference Systems Non-inertial reference systems are those that are accelerating. In these frames, Newton's laws do not hold without modification. * **Fictitious Forces:** To explain observed motion in non-inertial frames, "fictitious" or "inertial" forces are introduced. These are not real forces arising from physical interactions but are mathematical artifacts of the accelerating reference frame. Examples include centrifugal force and Coriolis force. > **Example:** In a car accelerating forward, you feel pushed back into your seat. This "force" is not a real interaction but an effect of your inertia resisting the acceleration of the car. ### 2.6 Uniform Circular Motion Uniform circular motion is motion in a circle at a constant speed. Although the speed is constant, the velocity is not, as the direction is continuously changing. * **Centripetal Acceleration:** This is the acceleration directed towards the center of the circle that is required to change the direction of the velocity. $$ a_c = \frac{v^2}{r} $$ where $v$ is the speed and $r$ is the radius of the circle. ### 2.7 Frictional Forces Friction is a force that opposes motion or intended motion between surfaces in contact. * **Static Friction:** The force that prevents an object from starting to move. It has a maximum value that must be overcome for motion to begin. $$ f_{s, \text{max}} = \mu_s N $$ where $\mu_s$ is the coefficient of static friction and $N$ is the normal force. * **Kinetic Friction:** The force that opposes motion when an object is already moving. It is typically constant and less than the maximum static friction. $$ f_k = \mu_k N $$ where $\mu_k$ is the coefficient of kinetic friction. * **Drag Force:** When an object moves through a fluid (like air or water), it experiences a drag force that opposes its motion. This force depends on the object's shape, speed, and the properties of the fluid. ### 2.8 Newton's Laws of Dynamics Newton's laws form the basis of classical mechanics. * **First Law (Law of Inertia):** An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. * **Second Law:** The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. $$ \vec{F}_{\text{net}} = m\vec{a} $$ where $\vec{F}_{\text{net}}$ is the net force, $m$ is the mass (inertial mass), and $\vec{a}$ is the acceleration. * **Third Law:** For every action, there is an equal and opposite reaction. When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body. > **Tip:** The first law is a special case of the second law where the net force is zero, resulting in zero acceleration (constant velocity). ### 2.9 Motion in Two and Three Dimensions Extending the concepts of velocity and acceleration to higher dimensions allows for the description of complex motion paths. * **Position, Velocity, and Acceleration Vectors:** These quantities are described by vectors with components in each dimension (e.g., $x$, $y$ in 2D; $x$, $y$, $z$ in 3D). * **Independence of Motion:** In many cases, motion in different dimensions can be analyzed independently (e.g., horizontal and vertical components of projectile motion). ### 2.10 General Motion Description The fundamental kinematic definitions of velocity and acceleration as derivatives of position apply to motion in any number of dimensions. The concept of using components allows complex motion to be broken down into simpler, one-dimensional analyses. --- # Newton's laws of motion and reference systems This section details Newton's three laws of dynamics, introducing inertial and non-inertial reference systems and the associated concepts of relative motion and fictitious forces. ### 3.1 Newton's three laws of dynamics Newton's laws of motion are fundamental principles that describe the relationship between an object's motion and the forces acting upon it. #### 3.1.1 The first law: Law of inertia The first law states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external, unbalanced force. This means that to change an object's velocity, which includes its speed or direction, a force must be applied. In everyday life, we observe objects slowing down due to frictional forces, which are external forces opposing motion. > **Tip:** The first law is a special case of the second law where the net force is zero. #### 3.1.2 The second law: Force and acceleration The 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. This mass, $m$, is referred to as inertial mass, representing the object's resistance to acceleration. The mathematical representation of this law is: $$ \vec{F}_{\text{net}} = m\vec{a} $$ Where: - $ \vec{F}_{\text{net}} $ is the net force acting on the object. - $ m $ is the mass of the object. - $ \vec{a} $ is the acceleration of the object. The concept of weight ($ \vec{W} $) is also introduced, which is the force needed to prevent a body from falling under the influence of gravity. #### 3.1.3 The third law: Action-reaction The third law states that for every action, there is an equal and opposite reaction. When two bodies interact, the force exerted by the first body on the second is equal in magnitude and opposite in direction to the force exerted by the second body on the first. > **Tip:** This law applies to all types of forces, including contact forces and forces acting at a distance. ### 3.2 Reference systems A reference system is a framework used to describe the motion of an object. The validity of Newton's laws depends on the choice of reference system. #### 3.2.1 Inertial reference systems An inertial reference system (IS) is a reference frame in which Newton's laws of motion hold true. Specifically, an object at rest stays at rest, and an object in motion continues in motion with constant velocity, unless acted upon by a net external force. Key characteristics of inertial reference systems: - They are either at rest or moving with a constant velocity relative to each other. - The laws of physics are the same in all inertial reference systems. - The first law of motion (law of inertia) is intrinsically defined by the concept of an inertial system. #### 3.2.2 Non-inertial reference systems A non-inertial reference system (NIS) is a reference frame that is accelerating. In such systems, Newton's laws do not hold in their standard form because of the acceleration of the reference frame itself. To describe motion in a non-inertial frame, fictitious forces must be introduced. - **Fictitious forces:** These forces are not caused by any physical interaction but are a consequence of the acceleration of the reference frame. They appear to act on objects within the non-inertial frame to explain observed accelerations. For example, in a reference frame accelerating with a trolley, an observer might invoke a fictitious force to explain why a ball appears to accelerate relative to them, even though no real force from physical interaction is acting on it in that frame. #### 3.2.3 Relative motion Relative motion describes how the observed motion of an object depends on the reference frame of the observer. When changing reference systems, velocities need to be adjusted accordingly. For instance, if an observer is in a moving vehicle, the velocity of an object outside will be the vector difference between the object's velocity and the vehicle's velocity. > **Example:** If a person throws a ball forward inside a moving train, and the train is moving with a velocity $ \vec{v}_{\text{train}} $ relative to the ground, and the ball is thrown with a velocity $ \vec{v}_{\text{ball, train}} $ relative to the train, then the ball's velocity relative to the ground is $ \vec{v}_{\text{ball, ground}} = \vec{v}_{\text{ball, train}} + \vec{v}_{\text{train}} $. #### 3.2.4 Uniform circular motion Uniform circular motion, where an object moves in a circle at a constant speed, is a crucial example in understanding forces. Although the speed is constant, the direction of velocity is continuously changing, meaning there is an acceleration. This acceleration, known as centripetal acceleration, is always directed towards the center of the circle. It requires a net force, the centripetal force, to maintain this motion. ### 3.3 Specific applications and concepts #### 3.3.1 Free-falling bodies Free fall occurs when an object is under the sole influence of gravity. The acceleration due to gravity, $g$, is independent of the object's mass, as demonstrated by dimensional analysis. The equations of motion for free fall are derived by integrating acceleration with respect to time: - Velocity as a function of time: $ v(t) = -gt + v_0 $ - Position as a function of time: $ y(t) = -\frac{1}{2}gt^2 + v_0t + h $ Where $ v_0 $ is the initial velocity and $ h $ is the initial height. If the object starts from rest at height $ h $, then $ v_0 = 0 $ and the equations simplify. #### 3.3.2 Projectile motion Projectile motion can be analyzed by considering the horizontal and vertical components of motion separately. An object thrown with an initial velocity $ \vec{V}_0 $ at an angle $ \alpha $ is subject to gravitational acceleration $ \vec{g} $. The motion can be described in two dimensions. The time of flight is determined by when the vertical position returns to zero. Projectile motion exhibits symmetry. #### 3.3.3 Frictional forces Frictional forces oppose motion or impending motion between surfaces in contact. - **Static friction:** This force prevents an object from starting to move. It has a maximum value that must be overcome for motion to begin. If a force is applied and the object doesn't move, the static friction force is equal in magnitude and opposite in direction to the applied force ($ \vec{F}_{\text{friction}} = -\vec{F}_{\text{applied}} $). - **Kinetic friction:** This force opposes the motion of an object that is already moving. It is generally less than the maximum static friction. The magnitude of frictional forces depends on the nature of the surfaces in contact and the normal force between them. #### 3.3.4 Drag force When an object moves through a fluid (liquid or gas), it experiences a drag force that opposes its motion. This force arises from turbulence in the fluid and depends significantly on the object's shape, speed, and the properties of the fluid. Aerodynamics is the study of drag forces. --- # Work, energy, and momentum This section explores the fundamental concepts of work, energy, and momentum, outlining their definitions, relationships, and the principle of conservation. ### 4.1 Work and energy #### 4.1.1 Definition of work Work is a scalar quantity that quantifies the energy transferred to or from an object by a force acting on it. It is related to the capacity of a body to do work. #### 4.1.2 Kinetic energy Kinetic energy ($KE$) is the energy an object possesses due to its motion. For a variable force, the calculation of work accounts for the non-constant nature of the force. #### 4.1.3 Work done by a force The work done by a force is calculated as the integral of the force over the displacement. If the force is constant and in the direction of displacement, work is given by $W = Fd$. For a force that is not constant or not in the direction of displacement, the work done is: $$W = \int \mathbf{F} \cdot d\mathbf{r}$$ When considering a force acting in three dimensions, the work done is the scalar product of the force vector and the displacement vector. #### 4.1.4 Power Power is defined as the rate at which work is done or energy is transferred per unit of time. $$P = \frac{dW}{dt}$$ The unit of power is the watt (W), which is equal to one joule per second. #### 4.1.5 Work done by gravitational force The work done by the gravitational force on an object moving vertically is given by $W_g = -mg\Delta y$, where $m$ is the mass, $g$ is the acceleration due to gravity, and $\Delta y$ is the change in vertical position. #### 4.1.6 Potential energy Potential energy ($PE$) is the energy an object possesses due to its position or configuration. * **Gravitational potential energy:** For an object near the Earth's surface, the gravitational potential energy is given by $PE_g = mgy$, where $y$ is the height above a reference point. * **Gravitational potential:** This is a property of the gravitational field itself, defined as the potential energy per unit mass. #### 4.1.7 Conservative forces Conservative forces are forces for which the work done in moving an object between two points is independent of the path taken. For these forces, potential energy can be defined. Gravity is a prime example of a conservative force. #### 4.1.8 Conservation of energy The principle of conservation of mechanical energy states that in an isolated system where only conservative forces are doing work, the total mechanical energy (the sum of kinetic and potential energy) remains constant. $$E_{total} = KE + PE = constant$$ If external forces or non-conservative forces (like friction) are present, the total energy of the system may change, but the total energy of the universe is always conserved. #### 4.1.9 Escape velocity Escape velocity is the minimum speed an object needs to escape the gravitational pull of a massive body, assuming no other forces act upon it. For a body to escape Earth's gravity, its kinetic energy must be at least equal to the magnitude of its gravitational potential energy. $$v_{escape} = \sqrt{\frac{2GM}{R}}$$ where $G$ is the gravitational constant, $M$ is the mass of the Earth, and $R$ is the radius of the Earth. ### 4.2 Center of mass and momentum #### 4.2.1 Center of mass The center of mass ($CM$) of a system of particles or a body is a point that moves as though all the system's mass were concentrated at that point and all external forces were applied to that point. For a system of $n$ particles, the coordinates of the center of mass are: $$x_{CM} = \frac{\sum_{i=1}^{n} m_i x_i}{\sum_{i=1}^{n} m_i}$$ $$y_{CM} = \frac{\sum_{i=1}^{n} m_i y_i}{\sum_{i=1}^{n} m_i}$$ $$z_{CM} = \frac{\sum_{i=1}^{n} m_i z_i}{\sum_{i=1}^{n} m_i}$$ where $m_i$ is the mass of the $i$-th particle and $(x_i, y_i, z_i)$ are its coordinates. #### 4.2.2 Linear momentum Linear momentum ($\mathbf{p}$) is a vector quantity defined as the product of an object's mass and its velocity. $$\mathbf{p} = m\mathbf{v}$$ Newton's second law can be expressed in terms of momentum as the rate of change of momentum: $$\mathbf{F}_{net} = \frac{d\mathbf{p}}{dt}$$ #### 4.2.3 Momentum conservation The principle of conservation of linear momentum states that if the net external force acting on a system is zero, the total linear momentum of the system remains constant. This is a fundamental principle, especially in understanding collisions. #### 4.2.4 Collisions Collisions involve interactions between objects over a short period. Momentum is always conserved in collisions. * **Elastic collisions:** In elastic collisions, both momentum and kinetic energy are conserved. Perfect elastic collisions are idealized and rarely occur in macroscopic systems due to energy dissipation (e.g., into heat or deformation). Examples include Rutherford scattering and collisions of ideal gas molecules. * **Inelastic collisions:** In inelastic collisions, momentum is conserved, but kinetic energy is not. Some kinetic energy is converted into other forms of energy, such as heat, sound, or deformation. A perfectly inelastic collision is one where the objects stick together after the collision. > **Tip:** While kinetic energy might not be conserved in inelastic collisions, the total energy of the system (including all forms of energy) is always conserved. > **Example:** Imagine two billiard balls colliding. This is often approximated as an elastic collision because the energy lost to deformation and sound is relatively small. In contrast, if a car crashes into a wall and crumples, it's a highly inelastic collision where significant kinetic energy is converted into heat and deformation. --- # Rotational motion and fluids This section delves into the principles of rotational motion and the fundamental properties of fluids, including density, pressure, and buoyancy. ### 5.1 Rotational variables Rotational motion describes the movement of an object around a fixed axis. Key variables are used to quantify this motion: * **Angular position ($\theta$):** The angle of an object's orientation relative to a reference line. It is typically measured in radians. * **Angular displacement ($\Delta\theta$):** The change in angular position. * **Angular velocity ($\omega$):** The rate of change of angular position. $$ \omega = \frac{d\theta}{dt} $$ The SI unit for angular velocity is radians per second (rad/s). * **Angular acceleration ($\alpha$):** The rate of change of angular velocity. $$ \alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2} $$ The SI unit for angular acceleration is radians per second squared (rad/s$^2$). #### 5.1.1 Relationship between linear and rotational variables For a point on a rotating object at a distance $r$ from the axis of rotation: * **Linear speed ($v$):** $v = r\omega$ * **Tangential acceleration ($a_t$):** $a_t = r\alpha$ * **Centripetal acceleration ($a_c$):** This is the acceleration directed towards the center of rotation, responsible for changing the direction of the velocity. For uniform circular motion, $a_c = \frac{v^2}{r} = r\omega^2$. #### 5.1.2 Moment of inertia ($I$) Moment of inertia is the rotational analog of mass, representing an object's resistance to changes in its rotational motion. It depends on the mass distribution of the object and the axis of rotation. * For a system of discrete particles: $$ I = \sum_i m_i r_i^2 $$ where $m_i$ is the mass of the $i$-th particle and $r_i$ is its distance from the axis of rotation. * For a continuous body: $$ I = \int r^2 dm $$ where $dm$ is an infinitesimal mass element and $r$ is its distance from the axis of rotation. > **Tip:** Moment of inertia is crucial for understanding rotational dynamics, similar to how mass is for linear dynamics. Objects with mass distributed further from the axis of rotation have larger moments of inertia. ### 5.2 Fluids Fluids are substances that can flow and take the shape of their container. This category includes liquids and gases. #### 5.2.1 Density ($\rho$) Density is defined as the mass per unit volume of a substance. $$ \rho = \frac{m}{V} $$ * SI unit: kilograms per cubic meter (kg/m$^3$). * Fluids, particularly liquids, are often considered incompressible, meaning their density remains relatively constant under normal pressure changes. #### 5.2.2 Pressure ($P$) Pressure is the force exerted per unit area. $$ P = \frac{F}{A} $$ * SI unit: Pascals (Pa), where 1 Pa = 1 N/m$^2$. * In fluids, pressure is exerted equally in all directions. * Pressure increases with depth in a fluid due to the weight of the fluid above. For a fluid at rest, the pressure difference between two depths is given by: $$ \Delta P = \rho g h $$ where $\rho$ is the fluid density, $g$ is the acceleration due to gravity, and $h$ is the depth difference. #### 5.2.3 Buoyancy Buoyancy is the upward force exerted by a fluid that opposes the weight of an immersed object. According to Archimedes' principle, this buoyant force is equal to the weight of the fluid displaced by the object. * **Buoyant force ($F_B$):** $$ F_B = \rho_{fluid} V_{displaced} g $$ where $\rho_{fluid}$ is the density of the fluid, $V_{displaced}$ is the volume of the fluid displaced by the object, and $g$ is the acceleration due to gravity. > **Example:** An object floats if its weight is less than or equal to the buoyant force when fully submerged. It sinks if its weight is greater than the buoyant force. This is why a large ship made of steel can float – its overall density (including the air inside its hull) is less than that of water. > **Example:** Scuba divers must breathe compressed air because the surrounding water exerts significant pressure. The compressed air allows their lungs to expand against this pressure. ### 5.3 Rotational dynamics and fluids examples While the provided text briefly mentions rotational variables in the context of calculating something related to speed, frequency, and radius (likely a tangential speed calculation), and then transitions to fluids, a detailed combination of rotational dynamics and fluid mechanics is not elaborated upon in the given pages. The text does, however, provide context for fluid properties and the concept of buoyancy. --- ## 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 | |------|------------| | Units | Standard quantities used to measure physical properties, such as meters for length or seconds for time. A physical quantity is meaningless without its associated units. | | Dimensional Analysis | A method used to check the consistency of physical equations by examining the dimensions of the quantities involved. It ensures that both sides of an equation have the same fundamental dimensions. | | Velocity | The rate of change of an object's position with respect to time, including both speed and direction. It is a vector quantity. | | Acceleration | The rate of change of an object's velocity with respect to time. It is also a vector quantity and can involve changes in speed, direction, or both. | | Free Fall | The motion of an object solely under the influence of gravity, neglecting air resistance. The acceleration due to gravity is constant near the Earth's surface. | | Projectile Motion | The motion of an object launched into the air, subject only to the force of gravity. It can be analyzed by considering independent horizontal and vertical components of motion. | | Newton's First Law of Motion | Also known as the law of inertia, it states that an object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. | | Newton's Second Law of Motion | States that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass, often expressed as $F=ma$. | | Newton's Third Law of Motion | 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. | | Inertial Reference System | A frame of reference in which Newton's laws of motion hold true. All such systems move with constant velocity relative to each other. | | Non-Inertial Reference System | A frame of reference that is accelerating. In these systems, fictitious forces must be introduced to explain observed motion according to Newton's laws. | | Work | In physics, work is done when a force causes displacement. It is calculated as the dot product of the force vector and the displacement vector, and its unit is the Joule (J). | | Kinetic Energy | The energy an object possesses due to its motion. It is calculated as $KE = \frac{1}{2}mv^2$, where $m$ is mass and $v$ is velocity. | | Potential Energy | Stored energy that an object has due to its position or state. Gravitational potential energy is an example, dependent on height and mass. | | Conservation of Energy | A fundamental principle stating that energy cannot be created or destroyed, only transformed from one form to another. The total energy of an isolated system remains constant. | | Linear Momentum | A measure of an object's motion, defined as the product of its mass and velocity ($p=mv$). It is a vector quantity. | | Momentum Conservation | In a closed system, the total linear momentum remains constant. This principle is crucial for analyzing collisions and interactions between objects. | | Elastic Collision | A collision in which both momentum and kinetic energy are conserved. No energy is lost or transformed into other forms. | | Inelastic Collision | A collision in which momentum is conserved, but kinetic energy is not. Some kinetic energy is typically lost and converted into heat, sound, or deformation. | | Density | A measure of mass per unit volume of a substance, calculated as $\rho = \frac{m}{V}$. It is an intrinsic property of materials. | | Pressure | The force exerted per unit area. In fluids, pressure increases with depth due to the weight of the fluid above. | | Buoyancy | The upward force exerted by a fluid that opposes the weight of an immersed object. It is equal to the weight of the fluid displaced by the object (Archimedes' principle). | | Center of Mass | The average position of all the mass in a system or object. It is the point at which the entire mass can be considered concentrated for translational motion calculations. |