AP Physics C Exam Review Notes

College Board Resources

• College board practice (index 1 to 33)
• Links to College Board videos:
  • https://www.youtube.com/watch?v=ZzwzMV9Nz8w&list=PLoGgviqq48467T6PU5a6QwxNgUb5SgI9V&index=1
  • https://www.youtube.com/watch?v=n3QYjaF8Zn0&list=PLoGgviqq48467T6PU5a6QwxNgUb5SgI9V&index=33

Exam Preparation Books

• The Princeton Review: AP Physics C Prep, 2023 (2 full-length practice tests).
• McGraw Hill: 5 Steps to a 5, AP Physics C, 2023 (3 practice exams).
• Barron's AP Physics C Premium, 2023 (4 full-length practice tests).

Guiding Principle: Motion Graphs

• Relationship between position, velocity, and acceleration graphs.
• Slope:
  • Position-time graph: slope gives velocity.
  • Velocity-time graph: slope gives acceleration.
• Area:
  • Velocity-time graph: area gives displacement.
  • Acceleration-time graph: area gives change in velocity.

Constant Acceleration Graphs (1D)

• Typical graphs for constant acceleration ($a ≠ 0$).
• For free fall, $a = g$ ($= 10 m/s^2$).

Constant Acceleration Motion Graphs ($a = 0$)

• Examples:
  • Toy cars (a > 0).
  • Free fall (a > 0).
  • Car on slope (a > 0).

Instantaneous Velocity (Calculus)

• Instantaneous velocity is the rate of change of position in time.
• $v = \frac{dx}{dt}$ (x and v are vectors).
• Instantaneous velocity is the slope (tangent) of the position-time graph at that time.

Guiding Principle: Projectile Motion

• Horizontal and vertical motions are independent.
• Horizontal motion: constant velocity.
• Vertical motion: constant acceleration g.

Projectile Motion Equations

• Horizontal equations can be derived from vertical equations by setting acceleration $g = 0$.
• Vertical equations:
  • $v<em>{yf} = v</em>{yi} + g\Delta t$
  • $\Delta y = \frac{1}{2}(v<em>{yi} + v</em>{yf})\Delta t$
  • $\Delta y = v_{yi}\Delta t + \frac{1}{2} g(\Delta t)^2$
  • $v<em>{yf}^2 = v</em>{yi}^2 + 2g\Delta y$
• Horizontal equations:
  • $v<em>{xf} = v</em>{xi}$
  • $\Delta x = v_{xi}\Delta t$

Projectile Motion: Resolve Components

• Resolve velocity into x and y components:
  • $v_x = v \cos{\Theta}$
  • $v_y = v \sin{\Theta}$

Guiding Principle: Problem Solving (Forces and Motion)

• Use the given motion to justify the force(s).
• Steps:
  • Identify objects.
  • Draw Free Body Diagram (FBD) of the object.
  • Identify contact forces + weight.
  • Solve the algebra.

Nature of Friction

• Static friction: keeps increasing until the object moves.
• Kinetic friction: stays the same when the object is moving.
• Kinetic friction is lower than the maximum static friction.

Guiding Principle: Circular Motion

• When an object is in circular motion, there must be a centripetal force.

Uniform Circular Motion

• Uniform circular motion means angular acceleration $α = 0$.
• Kinematics equations become:
  • $\omega<em>f = \omega</em>i$
  • $\Delta \theta = \omega_i(\Delta t)$
• These equations are similar to projectile motion equations in the x (horizontal) direction.

Guiding Principle: Work and Energy

• Work changes total mechanical energy E.
• $W = \Delta E$
  • Valid for systems with conservative forces.
• $E<em>{total} = K</em>{trans} + K<em>{rot} + U</em>g + U_s = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 + mgy + \frac{1}{2}kx^2$

System Perspectives: Energy

• Object-gravity system:
  • External force: gravity.
  • Work done by gravity: $W = mg\Delta y$.
  • GPE: none.
  • System energy: $E = K$.
  • Work-energy theorem: $W = \Delta E$
• Object + Earth system (object not in orbit):
  • External force: none.
  • Work done by gravity: none.
  • GPE: $mg\Delta y$.
  • System energy: $E = K + U_g$.
  • Work-energy theorem: $W = \Delta E = 0$
• Note: Potential energy (PE) requires an interaction between more than one object. A system of one object has no internal structure and cannot have PE.

Graph Representation: Energy

• Plot $F_{\parallel}$ vs. displacement.
• Work done by force = area under the graph.

Work Done by Perpendicular Force?

• Example: uniform circular motion without gravity.
• Instantaneous displacement $\Delta x$ is in the same direction as tangential speed v.
• Centripetal force is perpendicular to $\Delta x$ at any time.
• Tangential speed v is constant.
• Therefore $\Delta K = 0$.
• From work-energy principle, $W = \Delta K = 0$.
• Work done by a perpendicular force is zero.
• Perpendicular force only changes the direction of motion.

Work and Force in 2D (and 3D)

• $W = (F)(d)$ when the force is parallel to the displacement d.
• $W = 0$ when the force is perpendicular to the displacement d.
• Resolve a force into parallel and perpendicular components.
• Only the parallel component ($F \cos{\theta}$) does work.
• Therefore:
  • $W = Fd \cos{\theta}$ (F, d are + magnitudes).
  • $W = F<em>{\parallel}d$ ($F</em>{\parallel}$ can be + or -).
    • where $F_{\parallel} = F \cos{\theta}$.
  • $W = F \cdot d$ (vector dot product).

Comparison of Two Guiding Principles

• Impulse and Momentum:
  • $J = \Delta p$
  • $\Delta p = 0$ or p = constant.
• Work and Energy:
  • $W = \Delta E$
  • $\Delta E = 0$ or E = constant.

Energy Conservation

• Special case when no work is done.
• $\Delta E = \Delta K + \Delta U_g = 0$
• Therefore $E = K + U_g = constant$
• Total mechanical energy E is conserved.
• Total mechanical energy E is constant.

Momentum

• Force is change of momentum in time.
• $F = \frac{\Delta p}{\Delta t} = \frac{dp}{dt}$

Impulse and Force Graph

• $J = \Delta p$
• $F = \frac{dp}{dt}$ (slope formula).
• $J = \int F dt$ (area formula).

Problem Solving Strategy (Energy)

• Motion variables:
  • t, x, v, a
• Which motion variables are used in energy calculations? x, v
• Which motion variables are not used in energy calculations? t, a
• In general, see if you can solve the problem using energy. It is usually easier. If not, try Newton’s laws.

Comparison of Graphs: Slope Formula

• Acceleration (a):
  • $a = \frac{\Delta v}{\Delta t}$
  • x axis = time (t), y axis = velocity (v)
• Velocity (v):
  • $v = \frac{\Delta x}{\Delta t}$
  • x axis = time (t), y axis = position (x)
• Force (F):
  • $F = \frac{\Delta p}{\Delta t}$
  • x axis = time (t), y axis = momentum (p)
• Torque (\tau):
  • $\tau = \frac{\Delta L}{\Delta t}$
  • x axis = time (t), y axis = angular momentum (L)

Comparison of Graphs: Area Formula

• Impulse (J):
  • $J = F\Delta t$
  • x axis = time (t), y axis = force (F)
• Work (W):
  • $W = F\Delta x$
  • x axis = displacement (x), y axis = force (F)
• $\Delta p = F\Delta t$
  • x axis = time (t), y axis = force (F)
• $\Delta L = \tau \Delta t$
  • x axis = time (t), y axis = torque (\tau)

System Perspective (Momentum)

• A closed (isolated) system is one in which there is no net external force.
• An open system is one in which there is a net external force.

System Perspective (Energy)

• A closed (isolated) system is one in which there is no work done.
• An open system is one in which there is work done on/by the system.
• Path independence in closed systems.

Guiding Principle: Rotational Motion

• Use what you learned in linear motion.
• Skills and contents are similar.

Guessmology Table

• Linear motion vs. Rotational motion:
  • Displacement: x -> $\theta$
  • Velocity: v -> $\omega$
  • Acceleration: a -> $\alpha$
  • Inertia: m -> I
  • Newton’s 2nd law: $F = ma$ -> $\tau = I\alpha$
  • Work: $W = F_{\parallel}d$ -> $W = \tau \theta$
  • Kinetic energy: $K = \frac{1}{2} mv^2$ -> $K = \frac{1}{2} I \omega^2$
  • Momentum: $p = mv$ -> $L = I\omega$

Guessmology Table (cont.)

• Linear motion vs. Rotational motion:
  • Guiding principle unit 4: $W = \Delta E$ (same for both)
  • Guiding principle unit 5: $J = \Delta p$, $\Delta p = F \Delta t$ -> $\Delta L = \tau \Delta t$

Kinematics Equations Comparison

• Problem solving strategy:
  • Variables given?
  • Variables asked?
  • Choose equation (note: last equation not in equation sheet)
• What’s missing?
  • no displacement:
    • Rotation: $\omega<em>f = \omega</em>i + \alpha \Delta t$
    • Linear: $v<em>f = v</em>i + a\Delta t$
  • no final velocity:
    • Rotation: $\Delta \theta = \omega_i(\Delta t) + (\frac{1}{2})\alpha(\Delta t)^2$
    • Linear: $\Delta x = v_i(\Delta t) + (\frac{1}{2})a(\Delta t)^2$
  • no time:
    • Rotation: $\omega<em>f^2 = \omega</em>i^2 + 2\alpha(\Delta \theta)$
    • Linear: $v<em>f^2 = v</em>i^2 + 2a(\Delta x)$
  • no acceleration:
    • Rotation: $\Delta \theta = \frac{1}{2}(\omega<em>f + \omega</em>i)(\Delta t)$
    • Linear: $\Delta x = (\frac{1}{2})(v<em>f + v</em>i)(\Delta t)$

Keywords

• Linear Motion vs. Rotational Motion:
  • Linear displacement -> Angular displacement
  • Linear velocity -> Angular velocity
  • Linear acceleration -> Angular acceleration
  • Mass -> Rotational inertia
  • Work -> Work
  • Translational kinetic energy -> Rotational kinetic energy
  • Linear momentum -> Angular momentum

Rigid Object Equilibrium

• For point object in equilibrium: $\vec{F}_{net} = 0$
• For rigid object in equilibrium: $\tau<em>{net} = 0$ and $\vec{F}</em>{net} = 0$
• There is no rotation. You can choose the pivot point anywhere.

Rolling and Slipping (Sliding)

• Rolling means there is no slipping:
  • $s = r\theta$
  • $v_T = r\omega$
  • $a_T = r\alpha$
  • Static friction
• If there is slipping (sliding):
  • $s \neq r\theta$
  • $v_T \neq r\omega$
  • $a_T \neq r\alpha$
  • Kinetic friction

Guiding Principle: SHM

• A linear restoring force that is proportional to displacement.
• Restoring force $F = -kx$

Summary: SHM

• $x = A \cos(\omega t)$ position
• $v_x = -A\omega \sin(\omega t)$ velocity
• $a_x = -A\omega^2 \cos(\omega t)$ acceleration
• $a_x = -\omega^2 x$
• $\omega$ angular frequency
• A amplitude
• $f = \frac{\omega}{2\pi}$ frequency
• $T = \frac{1}{f}$ period

SHM (Calculus)

• $v = \frac{dx}{dt}$
• $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$
• $F = ma = m \frac{d^2x}{dt^2}$
• Linear restoring force $F = -kx$
• Substitute: $m \frac{d^2x}{dt^2} + kx = 0$
• Substitute $\frac{k}{m} = \omega^2$ (k and m are positive)
• $\frac{d^2x}{dt^2} + \omega^2 x = 0$ (2nd order differential equation in x)
• Solution:
  • $x = A \cos(\omega t + \varphi)$ or $x = a \cos(\omega t) + b \sin(\omega t)$
  • (A, $\varphi$) or (a, b) are constants of integration

Orbit Physics

• True for orbits around an infinite mass e.g.
  • Planet around Sun
  • Satellites around earth
• For planets and satellites in circular orbit
  • centripetal force = gravitational force
• For planets and satellites in elliptical orbit
  • Angular momentum is conserved
  • Energy is conserved