#AP Physics C: Mechanics - Unit 6: Oscillations Study Guide 🚀

Welcome to Unit 6! This unit is all about oscillations, which are back-and-forth motions you see everywhere. We'll cover simple harmonic motion (SHM), springs, pendulums, and even touch on wave motion. Think of it as a deep dive into repeating patterns of motion. Let's get started!

This unit makes up about 4-6% of the exam. While it's not the biggest topic, the concepts are crucial and often appear in combined problems. Spend time understanding the core ideas, especially SHM.

#Unit Overview

#The Big Question:

How do restoring forces predict and lead to harmonic motion? 💡

#Key Topics:

• Simple Harmonic Motion (SHM)
• Springs
• Pendulums
• Energy in SHM
• Resonance

#Exam Breakdown:

• Exam Weight: 4-6%
• Class Time: 4-7 class periods (45 minutes each)
• Practice: 20 MCQs and 2 FRQs in AP Classroom

#Key Vocabulary

• Simple Harmonic Motion (SHM): Motion where the restoring force is proportional to displacement and acts in the opposite direction. Think of it as a perfect back-and-forth motion.
• Oscillation: The back-and-forth movement of an object around its equilibrium position.
• Amplitude (A): The maximum displacement from the equilibrium position.
• Period (T): Time for one complete cycle of motion. Measured in seconds.
• Frequency (f): Number of cycles per second. Measured in Hertz (Hz). $f = \frac{1}{T}$
• Restoring Force: Force that pulls an object back to its equilibrium. Always points towards equilibrium.
• Spring Constant (k): Measure of a spring's stiffness. Higher k means a stiffer spring.
• Angular Frequency (ω): Rate of oscillation in radians per second. $ω = 2πf = \frac{2π}{T}$
• Phase Angle (φ): Initial angle of the oscillation at t=0. Determines the starting position.
• Resonance: When an object vibrates with maximum amplitude because of an external force at its natural frequency.

#Unit Questions to Ponder:

1. How does mass and spring constant affect the period of a mass-spring system?
2. How do string length and gravity affect the period of a simple pendulum?
3. How does energy change during SHM?
4. How does initial displacement affect the amplitude of SHM?
5. What is resonance and how does it occur?

#6.1 Simple Harmonic Motion (SHM)

#What is SHM?

Simple harmonic motion is a special type of periodic motion where the restoring force is directly proportional to the displacement and acts opposite to the displacement. This results in a smooth, sinusoidal motion. Think of a perfect swing or a mass bobbing up and down on a spring.

#The SHM Equation

The position of an object in SHM is described by:

$$x(t) = A \cos(ωt + φ)$$

Where:

• $x(t)$ is the position at time t
• $A$ is the amplitude
• $ω$ is the angular frequency
• $φ$ is the phase angle

#Springs

#Mass-Spring Systems

When a mass is attached to a spring and displaced from its equilibrium, it oscillates in SHM. The restoring force is provided by the spring. The period of oscillation depends on the mass (m) and the spring constant (k).

#Period of a Mass-Spring System

$$T = 2π\sqrt{\frac{m}{k}}$$

• T is the period (time for one oscillation)
• m is the mass
• k is the spring constant

#Pendulums

#Simple Pendulums

A simple pendulum consists of a mass (bob) attached to a string or rod, swinging back and forth. The restoring force is due to gravity. The period depends on the length of the string (L) and the acceleration due to gravity (g).

#Period of a Simple Pendulum

$$T = 2π\sqrt{\frac{L}{g}}$$

• T is the period
• L is the length of the pendulum
• g is the acceleration due to gravity (approximately 9.8 m/s² on Earth)

#Energy in SHM

#Energy Transformation

In SHM, energy is constantly transformed between kinetic energy (KE) and potential energy (PE). In a mass-spring system, PE is stored in the spring; in a pendulum, PE is gravitational. The total mechanical energy (E) of the system is conserved (assuming no friction).

#Energy Equations

• Maximum Potential Energy (Umax): $U_{max} = \frac{1}{2}kA^2$ (for springs)
• Maximum Kinetic Energy (Kmax): $K_{max} = \frac{1}{2}mv_{max}^2$
• Total Energy (E): $E = U_{max} = K_{max}$

#Resonance

#What is Resonance?

Resonance occurs when an external force is applied to an oscillating system at its natural frequency. This causes the amplitude of the oscillation to increase dramatically. Think of pushing a child on a swing at just the right time to make them go higher and higher.

#Practical Implications

Resonance can be both useful (e.g., in musical instruments) and dangerous (e.g., causing bridges to collapse). Understanding resonance is crucial in engineering and design.

#Practice Problems

Let's test your knowledge with some practice problems. These are designed to mimic the types of questions you'll see on the AP exam.

#Practice Questions:

1. A mass of 0.2 kg is attached to a spring with a spring constant of 20 N/m. What is the period of oscillation of the mass-spring system?
2. A pendulum of length 1 m is displaced from its equilibrium position by an angle of 10 degrees. What is the period of oscillation of the pendulum?
3. A mass-spring system has an amplitude of 5 cm and a period of 0.4 s. What is the maximum velocity of the mass during oscillation?
4. A pendulum has a period of 2 s on Earth. What would be its period on the moon, where the acceleration due to gravity is one-sixth of that on Earth?
5. A mass of 0.5 kg is attached to a spring with a spring constant of 10 N/m. If the mass is initially displaced by 0.2 m from its equilibrium position, what is its maximum potential energy during oscillation?

#Answers:

1. The period of oscillation of a mass-spring system is given by $T = 2π\sqrt{\frac{m}{k}}$, where m is the mass of the object and k is the spring constant. Plugging in the values, we get $T = 2π\sqrt{\frac{0.2}{20}} = 0.628$ s.
2. The period of oscillation of a simple pendulum is given by $T = 2π\sqrt{\frac{l}{g}}$, where l is the length of the string and g is the acceleration due to gravity. Plugging in the values, we get $T = 2π\sqrt{\frac{1}{9.81}} = 2.006$ s.
3. The maximum velocity of a mass-spring system is given by $v_{max} = Aω$, where A is the amplitude and ω is the angular frequency ($ω = \frac{2π}{T}$). Plugging in the values, we get $v_{max} = (0.05 m) × (\frac{2π}{0.4 s}) = 0.785$ m/s.
4. The period of oscillation of a simple pendulum is independent of the mass of the pendulum and is only dependent on the length of the string and the acceleration due to gravity. The period of the pendulum on the moon would be $T_{moon} = 2π\sqrt{\frac{L}{g/6}} = \sqrt{6} * 2s = 4.899s$
5. The maximum potential energy of a mass-spring system is given by $U_{max} = \frac{1}{2}kA^2$, where k is the spring constant and A is the amplitude. Plugging in the values, we get $U_{max} = (\frac{1}{2}) × 10 × (0.2)^2 = 0.2$ J.

Practice Question

#Practice Questions

#Multiple Choice Questions

1. A simple pendulum has a period T on Earth. If the pendulum is taken to a planet where the acceleration due to gravity is four times that of Earth, what is the new period of the pendulum in terms of T? (A) 4T (B) 2T (C) T/2 (D) T/4

2. A mass-spring system oscillates with a period of 2 seconds. If the mass is doubled and the spring constant is halved, what is the new period of oscillation? (A) 1 s (B) 2 s (C) 4 s (D) 8 s

3. A block attached to a spring oscillates horizontally on a frictionless surface. At which point in its motion is the kinetic energy of the block the greatest? (A) At the maximum displacement from equilibrium. (B) At the equilibrium position. (C) When the block is momentarily at rest. (D) The kinetic energy is constant throughout the motion.

#Free Response Question

A 0.5 kg block is attached to a spring with a spring constant of 20 N/m. The block is initially displaced 0.1 m from its equilibrium position and released from rest.

(a) Calculate the period of oscillation of the block. (b) Calculate the maximum speed of the block during its oscillation. (c) Calculate the total energy of the system. (d) If the block is now placed on a surface with friction, describe how the amplitude of the oscillation changes over time.

#FRQ Scoring Breakdown

(a) Period of oscillation: - Correct formula: 1 point - Correct substitution: 1 point - Correct answer with units: 1 point

(b) Maximum speed: - Correct formula or concept: 1 point - Correct substitution: 1 point - Correct answer with units: 1 point

(c) Total energy: - Correct formula: 1 point - Correct substitution: 1 point - Correct answer with units: 1 point

(d) Effect of friction: - Correct description of decreasing amplitude: 1 point

#Final Exam Focus

#High-Priority Topics:

• Simple Harmonic Motion (SHM): Understand the conditions for SHM and the equations that describe it.
• Mass-Spring Systems: Be able to calculate the period, frequency, and energy of a mass-spring system.
• Simple Pendulums: Be able to calculate the period of a simple pendulum. Remember that the period is independent of mass.
• Energy in SHM: Understand the transformation between kinetic and potential energy in SHM.
• Resonance: Know the conditions for resonance and its implications.

#Common Question Types:

• Multiple Choice: Conceptual questions about SHM, energy transformations, and how changes in mass, spring constant, or length affect the period.
• Free Response: Problems involving calculations of period, frequency, velocity, and energy in mass-spring systems and pendulums. Often includes a part about energy conservation or the effect of friction.

#Last-Minute Tips:

• Time Management: Don't spend too long on any one question. If you're stuck, move on and come back later.
• Units: Always include units in your calculations and final answers.
• Formulas: Make sure you know the key formulas for SHM, mass-spring systems, and pendulums. Write them down at the start of the exam.
• Conceptual Understanding: Focus on understanding the concepts rather than just memorizing formulas. This will help you tackle unfamiliar problems.
• Practice: Do as many practice problems as possible. This will help you get comfortable with the types of questions you'll see on the exam.

You've got this! With a solid understanding of these concepts and some practice, you'll be well-prepared for the AP Physics C: Mechanics exam. Good luck! 🌟