AP Physics C: Mechanics Study Guide

1. Introduction to AP Physics C: Mechanics

AP Physics C: Mechanics focuses on the principles of motion, forces, energy, and momentum using calculus. The course is heavily reliant on problem-solving and applying physics concepts to real-world scenarios.

Exam Format:

• Multiple-Choice Questions: Assess conceptual understanding and problem-solving skills.
• Free-Response Questions: Require the application of physics principles to solve complex problems.

2. Kinematics

Displacement, Velocity, and Acceleration:

• Displacement is the change in position:
  • Δx = x₂ - x₁
• Velocity is the rate of change of displacement:
  • v = Δx / Δt
• Acceleration is the rate of change of velocity:
  • a = Δv / Δt

Equations of Motion: For constant acceleration:

• v = v₀ + at
• Δx = v₀t + ½at²
• v² = v₀² + 2aΔx

Where:

• v₀ is the initial velocity,
• v is the final velocity,
• a is acceleration,
• t is time,
• Δx is displacement.

3. Newton's Laws of Motion

First Law (Law of Inertia):
An object at rest stays at rest, and an object in motion stays in motion unless acted upon by an external force.

Second Law (F = ma):
The force acting on an object is equal to the mass of the object times its acceleration.

• F = ma
• Units: N = kg·m/s²

Third Law (Action and Reaction):
For every action, there is an equal and opposite reaction.

4. Work, Energy, and Power

Work (W):

• Work is done when a force is applied to an object, causing displacement:
  • W = F * d * cos(θ)
• Units: Joules (J) = N·m

Kinetic Energy (KE):

• The energy of motion is given by:
  • KE = ½mv²
• Where:
  • m is mass,
  • v is velocity.

Potential Energy (PE):

• Gravitational potential energy near Earth:
  • PE = mgh
• Where:
  • m is mass,
  • g is acceleration due to gravity,
  • h is height.

Work-Energy Theorem:

• The work done on an object is equal to the change in its kinetic energy:
  • W = ΔKE = KE_f - KE_i

Power (P):

• Power is the rate at which work is done:
  • P = W / t
• Units: Watts (W) = J/s

5. Momentum and Impulse

Momentum (p):

• Momentum is the product of an object's mass and velocity:
  • p = mv

Impulse (J):

• Impulse is the change in momentum:
  • J = Δp = FΔt
• Where:
  • F is the force,
  • Δt is the time interval.

Conservation of Momentum:

• In a closed system with no external forces, the total momentum is conserved:
  • p₁ + p₂ = constant

6. Rotational Motion

Angular Displacement, Velocity, and Acceleration:

• Angular displacement (θ), angular velocity (ω), and angular acceleration (α) are the rotational analogs to linear displacement, velocity, and acceleration.
• ω = Δθ / Δt
• α = Δω / Δt

Equations of Rotational Motion: For constant angular acceleration:

• ω = ω₀ + αt
• θ = θ₀ + ω₀t + ½αt²
• ω² = ω₀² + 2α(θ - θ₀)

7. Torque and Rotational Dynamics

Torque (τ):

• Torque is the rotational equivalent of force, causing an object to rotate:
  • τ = rF sin(θ)
• Where:
  • r is the distance from the axis of rotation,
  • F is the applied force,
  • θ is the angle between the force and the lever arm.

Moment of Inertia (I):

• The moment of inertia is the rotational equivalent of mass:
  • I = Σmr²
  • Where:
    • m is the mass of each particle,
    • r is the distance from the axis of rotation.

Rotational Kinetic Energy (K_rot):

• The rotational kinetic energy of a rotating object is given by:
  • K_rot = ½Iω²

8. Gravitation

Newton’s Law of Universal Gravitation:

• The gravitational force between two masses is:
  • F = G * (m₁m₂) / r²
  • Where:
    • G = 6.67 × 10⁻¹¹ N·m²/kg² is the gravitational constant,
    • m₁ and m₂ are the masses,
    • r is the distance between the centers of mass.

Gravitational Potential Energy:

• The gravitational potential energy between two masses is:
  • PE = -G * (m₁m₂) / r

Gravitational Field (g):

• The gravitational field is the force per unit mass:
  • g = G * m / r²
• The gravitational field at the surface of the Earth is approximately 9.8 m/s².

9. Simple Harmonic Motion

Hooke's Law:

• The restoring force for an object in simple harmonic motion (SHM) is proportional to the displacement from equilibrium:
  • F = -kx
• Where:
  • k is the spring constant,
  • x is the displacement from equilibrium.

Period and Frequency of SHM:

• The period (T) and frequency (f) are related to the mass (m) and the spring constant (k):
  • T = 2π√(m/k)
  • f = 1/T

Energy in SHM:

• The total mechanical energy in SHM is constant and is given by:
  • E = ½kA²
• Where:
  • A is the amplitude of the motion.

10. Conservation Laws

Conservation of Mechanical Energy:

• In the absence of non-conservative forces, the total mechanical energy (kinetic + potential) remains constant:
  • KE_i + PE_i = KE_f + PE_f

Conservation of Angular Momentum:

• The angular momentum of a system is conserved if no external torque acts on it:
  • L = Iω
  • Where:
    • L is the angular momentum,
    • I is the moment of inertia,
    • ω is the angular velocity.