#AP Physics C: Mechanics - Simple Harmonic Motion (SHM) Study Guide

Hey there! Let's get you prepped for the AP Physics C exam with a deep dive into Simple Harmonic Motion (SHM). This guide is designed to be your go-to resource, especially the night before the big day. We'll break down everything you need to know, keep it engaging, and make sure you're feeling confident. Let's do this!

#Introduction to Simple Harmonic Motion (SHM)

Simple Harmonic Motion (SHM) is all about oscillations that follow a smooth, repeating pattern—like a sine or cosine wave. It's super important because it shows up everywhere in physics, from springs to pendulums. We'll explore the key concepts, equations, and graphs you need to ace the exam.

SHM is a cornerstone of mechanics, frequently appearing in both multiple-choice and free-response questions. Mastery of SHM concepts is crucial for a high score.

#Displacement, Velocity, and Acceleration in SHM

#Equations for SHM Displacement

• The displacement of an object in SHM from its equilibrium position is described by:
  • $x = A \cos(2 \pi f t)$ or $x = A \sin(2 \pi f t)$
  • A is the amplitude (maximum displacement), f is the frequency, and t is time.
• Key Insight: Displacement, velocity, and acceleration are all interconnected in SHM. When displacement is at its max, velocity is zero, and vice versa. 🌊

#Differential Equation for SHM

• Newton's second law leads to a second-order differential equation that describes SHM:
  • $$\frac{d^2x}{dt^2} = -\omega^2 x$$
  • $\omega$ is the angular frequency.
• Solving this equation gives us the position function, which is a sinusoidal function.

#Characteristics from Position Equation

• Starting with the position equation $x = A \cos(\omega t + \phi)$:
  • $\phi$ is the phase constant.
• Acceleration: The acceleration in SHM is proportional to the displacement and always points towards the equilibrium:
  • $a = -\omega^2 x$ 🏃‍♂️
• Maximum Velocity: Occurs at the equilibrium position:
  • $v_{max} = A\omega$
• Maximum Acceleration: Occurs at the extremes of oscillation:
  • $a_{max} = A\omega^2$

#Resonance in Oscillating Systems

• Resonance happens when an external force is applied at the system's natural frequency.
• Natural Frequency is the frequency at which the system oscillates freely.
• At resonance, the amplitude of the oscillation increases dramatically. 📈

#Amplitude vs. Period in SHM

• The period of SHM is independent of the amplitude.
• Changing the amplitude doesn't change the time for one full cycle.
• The period depends only on the mass and the restoring force (e.g., spring stiffness or gravity in a pendulum).

#Graphical Analysis of SHM

• Graphs of displacement, velocity, and acceleration vs. time help analyze SHM.
• Displacement-time: Sinusoidal, amplitude is max displacement, period is one full cycle. 📊
• Velocity-time: Sinusoidal, $\frac{\pi}{2}$ (90°) phase shift relative to displacement.
• Acceleration-time: Sinusoidal, 180° out of phase with displacement.

#Final Exam Focus

Okay, let's zoom in on what's most crucial for the exam:

• High-Priority Topics: Equations of SHM, differential equation, relationships between displacement, velocity, and acceleration, resonance, and graphical analysis.
• Common Question Types: Identifying SHM, applying SHM equations to solve problems, interpreting graphs, and understanding resonance.
• Time Management: Practice solving problems quickly. Don't spend too long on one question. If you're stuck, move on and come back later.
• Common Pitfalls: Confusing amplitude and period, not understanding phase relationships, and misinterpreting graphs.

#Practice Questions

Practice Question

#Multiple Choice Questions

1. A mass on a spring oscillates with simple harmonic motion. If the amplitude of the motion is doubled, what happens to the total energy of the system? (A) It remains the same (B) It is doubled (C) It is quadrupled (D) It is halved

2. An object undergoing SHM has a maximum velocity of 2 m/s and a period of 4 s. What is the amplitude of the motion? (A) 0.64 m (B) 1.27 m (C) 2.55 m (D) 3.14 m

3. The displacement of an object in SHM is given by $x(t) = 5\cos(2\pi t)$. What is the acceleration of the object at $t = 0.5$ s? (A) 0 m/s² (B) 5 m/s² (C) -20\pi^2 m/s² (D) 20\pi^2 m/s²

#Free Response Question

A 0.5 kg block is attached to a spring with a spring constant of 200 N/m. The block is pulled 0.1 m from its equilibrium position and released from rest. Assume no friction.

(a) Calculate the angular frequency of the oscillation. (b) Determine the maximum velocity of the block. (c) Write an equation for the position of the block as a function of time, assuming the block is released at t = 0. (d) What is the total energy of the system? (e) If the same block is attached to a different spring with a spring constant of 800 N/m, how would the period of oscillation change? Justify your answer.

#Scoring Rubric

(a) (2 points)

• 1 point for using the correct formula: $\omega = \sqrt{\frac{k}{m}}$
• 1 point for correct answer: $\omega = \sqrt{\frac{200}{0.5}} = 20 \text{ rad/s}$

(b) (2 points)

• 1 point for using the correct formula: $v_{max} = A\omega$
• 1 point for correct answer: $v_{max} = (0.1)(20) = 2 \text{ m/s}$

(c) (3 points)

• 1 point for using the correct form: $x(t) = A\cos(\omega t + \phi)$
• 1 point for correct amplitude: $A = 0.1 \text{ m}$
• 1 point for correct phase constant (since it starts at max displacement): $\phi = 0$
  • Final answer: $x(t) = 0.1\cos(20t)$

(d) (2 points)

• 1 point for using the correct formula: $E = \frac{1}{2}kA^2$
• 1 point for correct answer: $E = \frac{1}{2}(200)(0.1)^2 = 1 \text{ J}$

(e) (3 points)

• 1 point for stating the period would decrease.
• 1 point for using the formula $T = 2\pi\sqrt{\frac{m}{k}}$ to justify.
• 1 point for explaining that increasing the spring constant decreases the period.

#Answers to Multiple Choice Questions

1. (C) It is quadrupled
2. (B) 1.27 m
3. (D) 20\pi^2 m/s²

Alright, you've got this! Remember to stay calm, trust your preparation, and tackle each question step by step. You're ready to rock this exam! 🚀