#AP Physics 1: Simple Harmonic Motion - Your Night-Before Guide

Hey there, future AP Physics master! Let's get you prepped for Simple Harmonic Motion (SHM). This guide is designed to be quick, clear, and super helpful for your last-minute review. Let's dive in!

#1. Introduction to Simple Harmonic Motion (SHM)

• Periodic Motion: Any motion that repeats itself at regular intervals.
• Key Idea: SHM happens when a restoring force pulls an object back to its equilibrium position. 💡

#1.1. What Makes SHM Special?

• Restoring Force: This force always points towards the equilibrium position. It's like an invisible string pulling the object back to its center.
• Proportionality: The restoring force is directly proportional to the object's displacement from equilibrium. This is the heart of SHM!

#2. Restoring Force and Displacement

#2.1. Hooke's Law: The SHM Equation

• Hooke's Law: Describes the relationship between restoring force ($F$) and displacement ($x$): $F = -kx$
  • $F$: Restoring force (N)
  • $k$: Spring constant (N/m) - a measure of the stiffness of the spring
  • $x$: Displacement from equilibrium (m)
  • The negative sign indicates that the force acts in the opposite direction to the displacement.

#2.2. Equilibrium Position

• Equilibrium: The position where the net force on the object is zero. ⚖️
• Zero Acceleration: At equilibrium, the object has no acceleration because there's no net force.
• The Pull Back: When displaced, the restoring force causes the object to accelerate back towards equilibrium.

#2.3. Examples of SHM

• Mass on a Spring: A classic example where the spring provides the restoring force.
• Swinging Pendulum: For small angles, the pendulum's motion approximates SHM.

#2.4. Pendulum and SHM

• Restoring Torque: For a pendulum, the restoring force is actually a torque ($\tau$) that tries to bring the pendulum back to its lowest point.
• Small Angle Approximation: For small angular displacements ($\theta$), the restoring torque is proportional to the angular displacement: $\tau = -mgL\theta$
  • $m$: mass of the pendulum bob (kg)
  • $g$: acceleration due to gravity (9.8 m/s²)
  • $L$: length of the pendulum (m)

#3. Key Concepts and Connections

• Energy Conservation: In SHM, energy is constantly exchanged between potential and kinetic energy.
• Amplitude: The maximum displacement from equilibrium.
• Period (T): The time for one complete oscillation.
• Frequency (f): The number of oscillations per second.

#4. Final Exam Focus

Focus on understanding the relationship between restoring force, displacement, and energy in SHM. These concepts are frequently tested!

#4.1. High-Priority Topics

• Hooke's Law: Be ready to apply it in various contexts.
• Energy in SHM: Understand how potential and kinetic energy transform during oscillations.
• Pendulum Motion: Know the small-angle approximation and how it relates to SHM.

#4.2. Common Question Types

• MCQs: Expect questions that test your understanding of the relationships between force, displacement, and energy.
• FRQs: Look for problems that involve analyzing graphs of SHM or applying Hooke's Law to calculate forces and displacements.

#4.3. Last-Minute Tips

• Time Management: Don't get bogged down on a single question. Move on and come back if you have time.
• Common Pitfalls: Watch out for sign errors when applying Hooke's Law. Remember that the restoring force always acts opposite to the displacement.
• Strategies: Draw free-body diagrams to visualize the forces acting on the object. Use energy conservation to solve problems involving SHM.

#5. Practice Questions

Practice Question

#Multiple Choice Questions

1. A mass attached to a spring oscillates with simple harmonic motion. At which point in its motion is the magnitude of the acceleration of the mass the greatest? (A) When the mass is at its equilibrium position. (B) When the mass is at its maximum displacement from the equilibrium position. (C) When the mass is moving with its maximum velocity. (D) When the mass is moving with half of its maximum velocity.

2. A pendulum is set into motion. Which of the following statements is true about the restoring force acting on the pendulum bob? (A) It is constant throughout the motion. (B) It is maximum at the equilibrium position. (C) It is zero at the maximum displacement. (D) It is proportional to the displacement from the equilibrium position for small angles.

#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. Assume no friction.

(a) Calculate the potential energy stored in the spring when the block is at its maximum displacement. (2 points)

(b) Determine the maximum speed of the block as it passes through the equilibrium position. (3 points)

(c) Calculate the period of the oscillation. (2 points)

(d) If the block is replaced with a 1.0 kg block, how would the period of oscillation change? Explain your reasoning. (3 points)

#Answer Key & Scoring Breakdown:

MCQ Answers:

1. (B)
2. (D)

FRQ Scoring:

(a) Potential Energy: $U = \frac{1}{2}kx^2$

• $U = \frac{1}{2}(200 N/m)(0.1 m)^2 = 1 J$ (1 point for formula, 1 point for answer)

(b) Maximum Speed: $K = U$ (Energy conservation)

• $\frac{1}{2}mv^2 = 1 J$
• $v = \sqrt{\frac{2J}{0.5 kg}} = 2 m/s$ (1 point for energy conservation, 1 point for correct substitution, 1 point for answer)

(c) Period: $T = 2\pi\sqrt{\frac{m}{k}}$

• $T = 2\pi\sqrt{\frac{0.5 kg}{200 N/m}} = 0.31 s$ (1 point for formula, 1 point for answer)

(d) Period with 1 kg block: $T_{new} = 2\pi\sqrt{\frac{1 kg}{200 N/m}} = 0.44 s$

• The period will increase. (1 point)
• The period is proportional to the square root of the mass. (1 point)
• Increasing the mass increases the period. (1 point)

Alright, you've got this! Review these concepts, take a deep breath, and go ace that AP Physics 1 exam! You're well-prepared and ready to shine. Good luck! ✨