Simple Harmonic Motion (SHM) Study Guide 🎢

Welcome to your ultimate guide for Simple Harmonic Motion! Let's break down this key topic to ensure you're fully prepared for the AP Physics 1 exam. Remember, SHM is all about back-and-forth motion around an equilibrium point. Let's dive in!

Jump to Displacement, Velocity, and Acceleration

Jump to Amplitude and Period

Jump to Graphical Analysis

Jump to Practice Questions

Understanding Simple Harmonic Motion (SHM)

Key Components of SHM

• Amplitude (A): The maximum displacement from the equilibrium position.
• Frequency (f): The number of oscillations per second, measured in Hertz (Hz).
• Period (T): The time it takes for one complete oscillation.

These elements allow us to predict the position, speed, and direction of motion at any given time.

SHM is a fundamental concept that often appears in various forms on the AP exam. It's crucial to understand the relationships between displacement, velocity, and acceleration.

Displacement, Velocity, and Acceleration in SHM

Equations for Displacement in SHM

The displacement of an object in SHM can be modeled using sinusoidal functions (sine or cosine):

$$x(t) = A \cos(2\pi f t)$$

Or,

$$x(t) = A \sin(2\pi f t)$$

Where:

• $x(t)$ is the displacement at time $t$
• $A$ is the amplitude (maximum displacement)
• $f$ is the frequency
• $t$ is the time elapsed

Key Relationships

• Minima, Maxima, and Zeros: SHM exhibits distinct minima (lowest points), maxima (highest points), and zeros (equilibrium positions) for displacement, velocity, and acceleration.
• Equilibrium Position: At the equilibrium position, displacement and acceleration are zero, while velocity is at its maximum.
• Maximum Displacement (Amplitude): At the maximum displacement, velocity is zero and acceleration is at its maximum in the opposite direction of displacement.

Examples

• Example 1: At $t=0$, if displacement is at its maximum (amplitude), then velocity is zero, and acceleration is at its maximum, pointing towards the equilibrium.
• Example 2: When displacement passes through equilibrium, velocity reaches its maximum, and acceleration momentarily becomes zero.

Amplitude and Period in SHM

Independence of Amplitude and Period

• Increasing or decreasing the amplitude will not affect the time it takes to complete one full oscillation (period).
• The period depends on the system's characteristics, such as mass and spring constant for a mass-spring system, or length for a pendulum.

Factors Affecting the Period

• Mass-Spring System: $T = 2\pi \sqrt{\frac{m}{k}}$ (where $m$ is mass and $k$ is the spring constant)
• Simple Pendulum: $T = 2\pi \sqrt{\frac{L}{g}}$ (where $L$ is the length of the pendulum and $g$ is the acceleration due to gravity)

Graphical Analysis of SHM

Displacement-Time Graphs

• Appear as sinusoidal curves.
• Amplitude equals the maximum displacement.
• Period equals the time for one complete cycle.

Velocity-Time Graphs

• Also appear sinusoidal, but shifted by $\frac{1}{4}$ period relative to displacement.
• Maximum velocity occurs when the object passes through equilibrium.

Acceleration-Time Graphs

• Sinusoidal and shifted by $\frac{1}{2}$ period relative to displacement.
• Maximum acceleration occurs at the maximum displacements (amplitude) and always points towards equilibrium.

Determining Properties from Graphs

• Period: Can be determined from any of these graphs by measuring the time for one complete cycle.
• Amplitude: The amplitude of displacement can be determined by measuring the maximum displacement from equilibrium on a displacement-time graph.

Caption: This image shows the displacement, velocity, and acceleration graphs for SHM. Notice the phase shifts between the curves.

Practice Questions

Practice Question

Multiple Choice Questions

1. A mass-spring system oscillates with a period T. If the mass is doubled, what is the new period? (A) T/2 (B) T/\sqrt{2} (C) T\sqrt{2} (D) 2T

2. At what point in its motion does an object in SHM have zero velocity and maximum acceleration? (A) Equilibrium position (B) Maximum displacement (C) Halfway between equilibrium and maximum displacement (D) At no point

3. Which of the following graphs represents the relationship between acceleration and displacement in SHM? (A) A straight line with a positive slope (B) A straight line with a negative slope (C) A parabola (D) A circle

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 the surface is frictionless.

(a) Calculate the period of oscillation.

(b) Determine the maximum speed of the block.

(c) Write an equation for the position of the block as a function of time, assuming the block starts at its maximum displacement.

(d) Sketch a graph of the block's acceleration as a function of time for one complete period.

Answer Key and Scoring Rubric

Multiple Choice Answers:

1. (C) $T = 2\pi \sqrt{\frac{m}{k}}$, so if $m$ is doubled, $T$ is multiplied by $\sqrt{2}$.
2. (B) At maximum displacement, velocity is zero and acceleration is maximum.
3. (B) $a = -\omega^2 x$, so acceleration is proportional to negative displacement.

Free Response Scoring Rubric:

(a) Period Calculation (3 points)

• 1 point for using the correct formula: $T = 2\pi \sqrt{\frac{m}{k}}$
• 1 point for correct substitution: $T = 2\pi \sqrt{\frac{0.5}{200}}$
• 1 point for correct answer: $T ≈ 0.314$ s

(b) Maximum Speed (3 points)

• 1 point for using the conservation of energy: $\frac{1}{2} kA^2 = \frac{1}{2} mv^2_{max}$
• 1 point for correct substitution: $\frac{1}{2} (200)(0.1)^2 = \frac{1}{2} (0.5)v^2_{max}$
• 1 point for correct answer: $v_{max} = 2$ m/s

(c) Position Equation (3 points)

• 1 point for recognizing the cosine function: $x(t) = A \cos(\omega t)$
• 1 point for correct angular frequency: $\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{200}{0.5}} = 20$ rad/s
• 1 point for correct equation: $x(t) = 0.1 \cos(20t)$

(d) Acceleration Graph (3 points)

• 1 point for sinusoidal shape
• 1 point for correct period
• 1 point for correct amplitude (maximum acceleration at $a = -\omega^2 A = -40 m/s^2$)

Final Exam Focus

Focus on understanding the relationships between displacement, velocity, and acceleration in SHM. Be prepared to analyze graphs and apply the formulas for period and frequency.

Highest Priority Topics

• Relationships: Displacement, velocity, and acceleration in SHM.
• Graphs: Analyzing displacement-time, velocity-time, and acceleration-time graphs.
• Formulas: Period of mass-spring systems and pendulums.

Common Question Types

• Multiple Choice: Conceptual questions about the phase relationships and energy in SHM.
• Free Response: Problems involving calculations of period, frequency, and maximum velocity, and analysis of SHM graphs.

Last-Minute Tips

• Time Management: Quickly identify the type of problem and apply the relevant formulas.
• Common Pitfalls: Avoid confusing the conditions at equilibrium and maximum displacement. Double-check your units.
• Strategies: Draw diagrams to visualize the motion. Practice with past AP questions.

Good luck, you've got this! 🚀