#AP Physics 1: Simple Harmonic Motion - The Night Before 🌃

Hey! Let's get you totally prepped for Simple Harmonic Motion (SHM). This guide is designed to be super clear, quick to use, and exactly what you need for a great score tomorrow.

#1. What is Simple Harmonic Motion?

Simple Harmonic Motion (SHM) happens when an object is pulled back to its equilibrium point by a force that's proportional to how far it is from that point. Think of it like a spring or a pendulum swinging back and forth. 💡

• Key Examples:
  • Mass on a spring (obeying Hooke's Law)
  • Pendulum (small angle displacement)

#2. Newton's Second Law & SHM

#Applying Newton's Second Law

Newton's second law, F = ma, is your go-to for analyzing motion. Here's how to use it with SHM:

1. Identify Forces: Figure out all the forces acting on the object (gravity, spring force, etc.).
2. Free-Body Diagram: Draw a diagram showing all the forces.
3. Mass & Acceleration: Note the object's mass and its acceleration.
4. F = ma Equation: Write the equation for the sum of forces: F = ma.
5. Solve for Acceleration: Plug in known values and solve for 'a'.
6. Velocity & Position: Use acceleration to find velocity (v = at) and position (x = at²/2).
7. Graph It: Visualize the motion by graphing velocity and position vs. time.
8. Solve Unknowns: Use equations to find things like the spring constant or initial displacement.
9. Repeat: If there are multiple objects, repeat the process for each.

#3. Restoring Force: The Heart of SHM

#What's a Restoring Force?

A restoring force is like a guide that pulls an object back to its happy place—its equilibrium position. This force is what causes the back-and-forth motion in SHM.

• Key Points:
  • It always pulls the object back to equilibrium.
  • Common in oscillating systems (pendulums, springs).
  • It's opposite to the object's displacement from equilibrium.
  • Can be caused by gravity, elastic forces, friction.
  • The strength is measured by the spring constant (k).
  • The force is calculated as F = -kx (where x is the displacement).

#Ideal Springs

For AP Physics 1, we're working with ideal springs. This means they follow Hooke's Law perfectly. (If you take AP C: Mech, this will change!)

#4. Period, Amplitude, and Frequency

#Key Definitions

• Amplitude: The maximum displacement from the equilibrium point.
• Period (T): Time for one full cycle of motion (in seconds). It's the inverse of frequency.
• Frequency (f): How many cycles per second (in Hertz, Hz). f = 1/T

#Period Equations

#Period of a Pendulum

$$T = 2\pi \sqrt{\frac{L}{g}}$$

Where:
* L = length of the pendulum * g = acceleration due to gravity

#Period of a Mass on a Spring

$$T = 2\pi \sqrt{\frac{m}{k}}$$

Where: * m = mass * k = spring constant

#Visualizing SHM

This graph shows the displacement of an object undergoing SHM over time. The amplitude is the maximum displacement from the equilibrium point.

This equation shows the relationship between period and angular velocity (ω). We'll dig into angular velocity more in Unit 7!

#5. Pendulums & Springs: Key Features

#Important Points to Remember

• Force & Acceleration: Always point in the same direction.
• Max Force & Acceleration: Occur at the extremes (fully stretched or compressed).
• Zero Velocity: At the extremes.
• Max Velocity: At the equilibrium point (where force and acceleration are zero).

#6. Example Problems

Example Problem 1:

A 1 kg mass is on a spring (k = 50 N/m). It oscillates vertically with no friction. The mass is initially displaced 0.2 m from equilibrium and released from rest. What is the period of the oscillation?

Solution:

Use the formula: T = 2π√(m/k)

T = 2 * 3.14 * √(1 kg / 50 N/m) = 0.89 seconds

Example Problem 2:

A 2 kg mass is on a spring (k = 100 N/m). It oscillates vertically with no friction. The mass is initially displaced 0.5 m from equilibrium and released from rest. What is the period of the oscillation?

Solution:

Use the formula: T = 2π√(m/k)

T = 2 * 3.14 * √(2 kg / 100 N/m) = 0.89 seconds

#7. Final Exam Focus 🎯

#High-Priority Topics

• Understanding SHM: What it is, and when it occurs.
• Restoring Force: How it works and the formula F = -kx.
• Period Equations: Pendulum and mass-spring systems.
• Energy Conservation: (Connecting to Unit 5) Energy transforms between kinetic and potential in SHM.
• Graphical Analysis: Interpreting position, velocity, and acceleration graphs.

SHM is often integrated with other topics like energy and forces. Be prepared to combine concepts!

#Common Question Types

• Multiple Choice: Conceptual questions about restoring forces, period, and energy.
• Free Response: Problems involving calculations of period, energy, and analyzing graphs.

#Last-Minute Tips

• Time Management: Don't spend too long on one question. Move on and come back if needed.
• Common Pitfalls: Double-check your units, and remember the negative sign in F = -kx.
• Strategy for Challenging Questions: Break down complex problems into smaller parts. Draw diagrams and write down knowns and unknowns.

#8. Practice Questions

Practice Question

#Multiple Choice Questions

1. A pendulum is set into motion. If the length of the pendulum is quadrupled, what happens to the period of the pendulum? (A) The period is halved. (B) The period is doubled. (C) The period is quadrupled. (D) The period remains the same.

2. A mass is attached to a spring and oscillates horizontally on a frictionless surface. At which point in the oscillation is the speed of the mass the greatest? (A) At the maximum displacement from equilibrium (B) At the equilibrium position (C) When the spring is fully compressed (D) When the spring is fully stretched

3. A spring with a spring constant k is stretched a distance x from its equilibrium position. What is the potential energy stored in the spring? (A) 1/2 * k * x (B) k * x (C) 1/2 * k * x^2 (D) k * x^2

#Free Response Question

A 0.5 kg block is attached to a spring with a spring constant of 200 N/m. The block is initially at rest at its equilibrium position on a horizontal frictionless surface. An external force then compresses the spring by 0.1 m and the block is released.

(a) Calculate the potential energy stored in the spring when it is compressed by 0.1 m.

(b) Calculate the maximum speed of the block after it is released.

(c) Calculate the period of oscillation of the block-spring system.

(d) If the block is replaced by a 1 kg block, how will the period of the oscillation change? Explain your reasoning.

#Scoring Guidelines

(a) Potential Energy (2 points)

• 1 point for using the correct formula for potential energy of a spring: U = 1/2 * k * x^2
• 1 point for correct substitution and answer: U = 1/2 * 200 N/m * (0.1 m)^2 = 1 J

(b) Maximum Speed (3 points)

• 1 point for recognizing that potential energy is converted to kinetic energy: 1/2 * m * v^2 = U
• 1 point for correct substitution: 1/2 * 0.5 kg * v^2 = 1 J
• 1 point for correct answer: v = 2 m/s

(c) Period of Oscillation (3 points)

• 1 point for using the correct formula for the period of a mass-spring system: T = 2π√(m/k)
• 1 point for correct substitution: T = 2π√(0.5 kg / 200 N/m)
• 1 point for correct answer: T ≈ 0.314 s

(d) Effect of Increased Mass (2 points)

• 1 point for stating that the period will increase.
• 1 point for explaining that period is proportional to the square root of mass: T = 2π√(m/k), so increasing the mass increases the period.

Good luck! You've got this! 💪