AP Physics C: Mechanics Comprehensive Study Guide

Prerequisites and Exam Fundamentals

Before engaging with AP Physics C: Mechanics, students must establish a foundational understanding of specific mathematical and conceptual prerequisites. Unlike AP Physics 1 and 2, which are algebra-based, the Physics C curriculum is entirely calculus-based. Proficiency in basic differentiation and integration is mandatory, as is the ability to solve separable differential equations. However, the calculus required is not as advanced as Calculus BC; for example, students will not encounter integration by parts. Furthermore, a prior understanding of high school level physics or AP Physics 1/2 is highly beneficial, as this course expands upon those classical mechanics concepts in significantly greater detail.

In terms of notation used throughout these notes, vector quantities are represented by variables in BOLD. This study guide also adopts the convention allowed by the College Board to treat the acceleration due to gravity near Earth's surface as $10\,m/s^2$, rather than the more precise $9.81\,m/s^2$. This simplification is permitted on the exam and generally does not cause significant deviance in calculated results.

Kinematics in One Dimension

Kinematics focuses on three key quantities: position ($x$), velocity ($v$), and acceleration ($a$). Within this unit, motion is analyzed strictly in one dimension. For scenarios where acceleration is constant and the relationship between velocity and position does not fluctuate relative to time or other variables, the three main kinematic equations are applied:

$x = x_0 + v_{x0}t + \frac{1}{2}a_xt^2$

$v_x = v_{x0} + a_xt$

$v_x^2 = v_{x0}^2 + 2a_x(x - x_0)$

If the velocity is constant, it is defined by the displacement over the time elapsed: $v_x = \frac{\Delta x}{\Delta t}$. Models for average velocity and average/constant acceleration are similarly structured:

$v_{x(avg)} = \frac{\Delta x}{\Delta t}$

$a_{x(avg)} = \frac{\Delta v_x}{\Delta t}$

Calculus-Based Motion Relationships

When acceleration is non-uniform or follows a specific function, the standard kinematic equations are insufficient, and calculus-based relationships must be employed. Velocity is defined as the first derivative of position with respect to time ($v_x = \frac{dx}{dt}$), and acceleration is the first derivative of velocity with respect to time ($a_x = \frac{dv_x}{dt}$). Using the fundamental theorem of calculus, we can determine displacement and change in velocity through integration:

$\Delta x = \int v_x \, dt$

$\Delta v = \int a_x \, dt$

These integrals physically represent the "area under the curve" on velocity-time or acceleration-time graphs.

Kinematic Problem Examples and Applications

Example 1: Free-Fall (Constant Acceleration)
Imagine dropping a basketball from a helicopter at a height of $500\,meters$ above Earth. To find the time ($t$) it takes to hit the ground, we identify the known variables: the initial velocity ($v_{x0}$) is $0\,m/s$, the acceleration ($a$) is $g = 10\,m/s^2$ downwards, and the displacement ($x$) is $500\,meters$. Using the equation $x = x_0 + v_{x0}t + \frac{1}{2}a_xt^2$ (where $x_0 = 0$):

$x = \frac{1}{2}a_xt^2 \rightarrow t = \sqrt{\frac{2x}{a}}$

$t = \sqrt{\frac{2(500)}{10}} = \sqrt{100} = 10\,s$

Example 2: Non-Uniform Velocity (Calculus Application)
Consider an experimental vehicle where velocity is expressed as a function of time: $v(t) = 2t^3$. To find how far the vehicle travels in $5\,seconds$ starting from rest, we utilize the integral relation:

$\Delta x = \int_{t_1}^{t_2} v(t) \, dt = \int_{0}^{5} 2t^3 \, dt$

$\Delta x = [\frac{1}{2}t^4]_0^5 = \frac{1}{2}(5^4) = 312.5\,meters$

Newton's Laws of Motion

Newton’s First Law
An object at rest or in uniform motion (constant velocity) will remain in that state unless acted upon by a net external force ($F_{net} = 0$). An object "at rest" is simply a subset of uniform motion where $v = 0$. This applies in all directions. For instance, a hockey puck on frictionless ice continues at a constant velocity because gravity and the normal force cancel out, resulting in zero net force. A spaceship in deep space also maintains constant velocity due to negligible external forces. Crucially, this law assumes the mass of the object remains constant.

Newton’s Second Law
The vector sum of forces acting on an object is proportional to its mass and its acceleration:

$\mathbf{F} = \sum \mathbf{F} = m\mathbf{a}$

Note that this formulation is a special case assuming constant mass.

Newton’s Third Law
For every action, there is an equal and opposite reaction. If object A exerts a force on object B, object B exerts a force of equal magnitude but opposite direction on object A:

$\mathbf{F}<em>{A \text{ on } B} = -\mathbf{F}</em>{B \text{ on } A}$

Action and reaction forces always act on different objects.

Forces, Equilibrium, and Friction

Forces are interactions involving pushes or pulls between two objects, categorized as contact forces (direct touch) or non-contact forces (acting at a distance). Equilibrium occurs when the net force on an object is zero. This is categorized as static equilibrium (object is not moving relative to the observer) or dynamic equilibrium (object is moving at a constant velocity).

Normal Force is the reaction force that prevents objects from passing through each other when in contact.

Friction Force opposes the relative sliding of two surfaces. It depends on the normal force ($N$) and the coefficient of friction ($\mu$). There are two types:

1. Static Friction ($f_s$): Occurs when there is no relative motion. It increases to match the applied force up to a maximum value: $f_s \le \mu_s N$.
2. Kinetic Friction ($f_k$): Occurs during relative motion and is constant if the normal force remains constant: $f_k = \mu_k N$.

Applications: Pulleys and Centripetal Motion

In a system involving two blocks and a frictionless pulley, you can determine acceleration by drawing free-body diagrams for each block and applying Newton’s Second Law. For a mass $M$ hanging and a mass $m$ on a table, the equations can be added to eliminate tension, resulting in:

$Mg = (M + m)a \rightarrow a = \frac{Mg}{M + m}$

Uniform Circular Motion
When an object travels in a circle at a constant speed, it experiences centripetal acceleration ($a_c$) pointing toward the center:

$a_c = \frac{v^2}{r}$

Centripetal force is not a standalone force; it is the label given to the net force (provided by tension, gravity, etc.) that maintains the circular path.

Work, Energy, and Power

Work ($W$) is done when a force displaces an object. It only accounts for the force component parallel to the displacement. For a constant force, $W = Fd$, where the unit is Joules ($1\,J = 1\,N\,m$). Using vectors and the dot product, work is defined as:

$W = \mathbf{F} \cdot \mathbf{r} = Fr\cos(\theta)$

For variable forces, work is the integral of force with respect to position, or the area under a force-position graph:

$W = \int_{x_1}^{x_2} \mathbf{F}(r) \cdot d\mathbf{r}$

The Work-Energy Theorem states that the net work done on a point mass equals its change in kinetic energy: $W_{net} = \Delta K$. Kinetic energy is defined as $K = \frac{1}{2}mv^2$.

Conservative vs. Dissipative Forces
A force is conservative if the work done depends only on the initial and final positions. The net work over a closed path is zero. Non-conservative (dissipative) forces, like friction, depend on the path taken. Potential energy ($U$) is related to conservative forces via:

$\Delta U = -\int_a^b \mathbf{F}_{cons} \cdot d\mathbf{r}$

In one dimension, the differential relationship is $F_x = -\frac{dU(x)}{dx}$.

Potential Energy and Conservation Laws

Elastic Potential Energy (Springs)
Hooke's Law defines the force of an ideal spring as $F_s = -k\Delta x$. Integrating this gives the potential energy:

$U_s = \frac{1}{2}k(\Delta x)^2$

For non-linear springs, such as one with the relationship $F_s = -k\Delta x^2$, the potential energy must be found by integration: $U_s = -\int -k\Delta x^2 \, dx = \frac{1}{3}k\Delta x^3$.

Gravitational Potential Energy
Near Earth, where elevation change is small, $F_g = mg$ and $U_g = mgh$. At non-trivial distances (e.g., satellites), the universal law is used: $F_G = \frac{Gm_1m_2}{r^2}$. The potential energy is:

$U_G = -\frac{Gm_1m_2}{r}$

Potential energy is zero at infinity.

Conservation of Mechanical Energy
In a conservative system (no external work), the total mechanical energy remains constant: $E = U + K$. If non-conservative forces like friction are present, the work they do changes the total energy: $W_{noncons} = \Delta E$.

Power
Power is the rate of change of energy:

$P = \frac{dE}{dt}$

$P_{avg} = \frac{W}{t} = \mathbf{F} \cdot \mathbf{v}$

Systems of Particles and Linear Momentum

Physics problems often simplify objects to their Center of Mass (CoM). For a collection of point masses, the CoM position is:

$x_{cm} = \frac{\sum m_ix_i}{\sum m_i}$

Using calculus for continuous objects: $x_{cm} = \frac{\int x \, dm}{\int dm}$. If no net external force acts on a system, the CoM will not accelerate. If an object explodes in mid-air, the CoM of the fragments continues along the original trajectory.

Linear Momentum ($p$) is defined as $p = mv$. Newton's Second Law can be rewritten as the rate of change of momentum: $F = \frac{dp}{dt}$.

Impulse ($J$) is the average force over a time interval, equal to the change in momentum and the area under a force-time graph:

$J = F_{avg}\Delta t = \int F \, dt = \Delta p$

Collisions
Linear momentum is conserved in systems where no external forces act.

1. Elastic Collisions: Both momentum and kinetic energy are conserved.
2. Inelastic Collisions: Momentum is conserved, but kinetic energy is lost (e.g., to sound or heat).
3. Completely Inelastic Collisions: The objects stick together, resulting in the maximum possible loss of kinetic energy.

Rotational Dynamics

Rotation uses angular equivalents for displacement ($\theta$), velocity ($\omega = \frac{d\theta}{dt}$, in $rad/s$), and acceleration ($\alpha = \frac{d\omega}{dt}$). Tangential relationships include:

$v = r\omega$ AND $a_{\theta} = r\alpha$

For constant angular acceleration, rotational kinematics equations mirror linear ones, such as $\theta = \theta_0 + \omega_0t + \frac{1}{2}\alpha t^2$.

Torque ($\tau$) is the rotational effect of a force: $\tau = rF\sin(\phi)$, or the cross product $\mathbf{\tau} = \mathbf{r} \times \mathbf{F}$. The rotational version of Newton's Second Law is $\tau = I\alpha$, where $I$ is the rotational inertia.

Rotational Inertia (Moment of Inertia) is defined as $I = \sum r_i^2m_i$ for particles or $I = \int r^2 \, dm$ for continuous mass. It is the rotational equivalent of mass.

Angular Momentum ($L$) is $\mathbf{L} = \mathbf{r} \times \mathbf{p} = I\omega$. Its rate of change is torque: $\tau = \frac{dL}{dt}$.

Rotational Kinetic Energy is $K_{rot} = \frac{1}{2}I\omega^2$. Work by torque is $W = \int \tau \, d\theta$. The total kinetic energy of a rolling object is the sum of its translational and rotational kinetic energies.

Rolling without Slipping occurs when the contact point is stationary relative to the ground ($v = r\omega$ and $a = r\alpha$). This allows for conservation of energy calculations like: $mgh = \frac{1}{2}mv^2 + \frac{1}{2}I(\frac{v}{r})^2$.

Oscillations and SHM

Simple Harmonic Motion (SHM) is exemplified by a mass on a spring. A restoring force (Hooke's Law $F_s = -kx$) creates oscillation. The position function is:

$x(t) = A\cos(\omega t - \phi)$

Velocity and acceleration are derived via calculus:

$v(t) = -A\omega\sin(\omega t - \phi)$

$a(t) = -A\omega^2\cos(\omega t - \phi) = -\omega^2x$

Equating Newton's Second Law ($ma = -kx$) leads to the second-order differential equation: $\frac{d^2x}{dt^2} + \frac{k}{m}x = 0$. From this, angular frequency is determined as $\omega = \sqrt{\frac{k}{m}}$.

Pendulums

• Simple Pendulum: For small angles, $\omega = \sqrt{\frac{g}{l}}$.
• Physical Pendulum: Objects with rotational inertia pivoting at distance $L_{cm}$ from the CoM: $\omega = \sqrt{\frac{mgL_{cm}}{I}}$.

Period and Energy
The period ($T$) is the reciprocal of frequency ($f$): $T = \frac{2\pi}{\omega} = \frac{1}{f}$. For a spring-mass system, $T = 2\pi\sqrt{\frac{m}{k}}$. The total energy in SHM is a constant sum of kinetic and potential energy: $E_{total} = \frac{1}{2}kA^2$.

Gravitation and Orbits

Newton's Universal Law of Gravitation defines the force between two masses:

$F_{12} = G\frac{m_1m_2}{|r_{12}|^2}\hat{r}_{12} \text{ with magnitude } |F_G| = \frac{Gm_1m_2}{r^2}$

$G \approx 6.674 \times 10^{-11} \, Nm^2/kg^2$. For non-point masses, the force is the integral: $\mathbf{F} = GM \int \frac{dm}{r^2}\hat{r}$.

Gravitational Field ($g$) is a vector field: $\mathbf{g} = -\frac{Gm}{|r|^2}\hat{r}$. Close to Earth, this results in the constant $9.81\,m/s^2$. Net gravitational field is the vector sum of all fields present.

Orbital Dynamics
In a circular orbit, gravity provides the centripetal force ($\frac{GMm}{r^2} = \frac{mv^2}{r}$), leading to orbital velocity: $v_{orbit} = \sqrt{\frac{GM}{r}}$. Kepler's Third Law relates the period to the radius: $T^2 = \frac{4\pi^2r^3}{GM}$.

Finally, in any orbit, both Angular Momentum and Mechanical Energy are Conserved. Because gravity is a centripetal, conservative force, it does not affect angular momentum (which requires tangential force) and maintains a constant total sum of potential and kinetic energy.