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{yf} = v{yi} + g\Delta t
  • \Delta y = \frac{1}{2}(v{yi} + v{yf})\Delta t
  • \Delta y = v_{yi}\Delta t + \frac{1}{2} g(\Delta t)^2
  • v{yf}^2 = v{yi}^2 + 2g\Delta y
• Horizontal equations:
  • v{xf} = v{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:
  • \omegaf = \omegai
  • \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{total} = K{trans} + K{rot} + Ug + 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{\parallel}d (F{\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: \omegaf = \omegai + \alpha \Delta t
    • Linear: vf = vi + 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: \omegaf^2 = \omegai^2 + 2\alpha(\Delta \theta)
    • Linear: vf^2 = vi^2 + 2a(\Delta x)
  • no acceleration:
    • Rotation: \Delta \theta = \frac{1}{2}(\omegaf + \omegai)(\Delta t)
    • Linear: \Delta x = (\frac{1}{2})(vf + vi)(\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{net} = 0 and \vec{F}{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