#AP Physics 1: Energy in Simple Harmonic Motion (SHM) 🚀

Hey there, future physics ace! Let's get you prepped for the exam with a super-focused review of energy in SHM. We're going to break it down, make it stick, and get you feeling confident. Let's dive in!

#1. Conservation of Energy

#1.1 Internal Energy

• Definition: Internal energy (U) is the energy associated with the random motion of a system's particles. Think of it as the thermal energy within an object.
• SHM Connection: In a simple harmonic oscillator (like a mass on a spring), internal energy is stored as elastic potential energy when the oscillator is displaced from its equilibrium position.
• Energy Conversion: As the oscillator moves, internal energy is continuously converted between kinetic and potential energy.
• Periodicity: Internal energy in SHM follows a repeating pattern. It's maximum at maximum displacement and minimum at equilibrium.
• Formula: $U = \frac{1}{2}kx^2$ where:
  • U = Internal energy
  • k = Spring constant
  • x = Displacement from equilibrium

#1.2 Potential Energy

• Definition: Potential energy (U) is the energy an object has due to its position or configuration within a force field. It's the energy waiting to be released.
• SHM Connection: In SHM, potential energy is stored as elastic potential energy when the oscillator is displaced from its equilibrium position. It's due to the deformation of the spring or force-generating element.
• Periodicity: Like internal energy, potential energy is periodic, reaching its maximum at maximum displacement and minimum at equilibrium.
• Formula: $U = \frac{1}{2}kx^2$ (same as internal energy for a spring system)

#1.3 Kinetic Energy

• Definition: Kinetic energy (K) is the energy of motion. It's the energy an object has because it's moving.
• SHM Connection: In SHM, kinetic energy is the energy of the oscillator as it moves back and forth.
• Periodicity: Kinetic energy is periodic, reaching its maximum when the oscillator has maximum velocity and minimum at equilibrium or maximum displacement.
• Formula: $K = \frac{1}{2}mv^2$ where:
  • K = Kinetic energy
  • m = Mass
  • v = Velocity

#2. Energy in Simple Harmonic Oscillators

This section is HUGE for the AP exam! Expect to see questions combining energy and SHM concepts. Remember, it's all about energy transformation between potential and kinetic.

• Core Idea: Energy continuously converts between potential and kinetic forms during SHM. 🔄
• Maximum Potential Energy: Occurs when the spring is stretched or compressed the most (maximum displacement).
• Maximum Kinetic Energy: Occurs at the equilibrium point (zero displacement).

#2.1 Visualizing Energy in SHM

Here's a breakdown of how energy changes in a mass-spring system:

• A, C, E: Maximum potential energy, zero kinetic energy.
• B, D: Maximum kinetic energy, zero potential energy.

#2.2 Energy vs. Time Graph

• Total Energy (E) is Constant: No external forces are doing work on the system. The total mechanical energy is conserved (E = U + K).
• Parabolic Curves: The potential and kinetic energy graphs are curves due to the squared term in the potential energy equation ($U = \frac{1}{2}kx^2$).
• Phase Relationship: Potential energy is greatest when the position is at its maximum, and kinetic energy is greatest when the velocity is at its maximum.

#3. Example Problems

Let's solidify these concepts with a couple of example problems.

#3.1 Example Problem 1

Problem: A 1 kg mass is attached to a spring (k = 50 N/m). It's displaced 0.2 m from equilibrium and released from rest. What's the total energy at maximum displacement?

Solution:

• At maximum displacement, all energy is potential, and kinetic energy is zero.
• $U = \frac{1}{2}kx^2 = \frac{1}{2}(50 \frac{N}{m})(0.2 m)^2 = 1 J$
• Total Energy = 1 J

#3.2 Example Problem 2

Problem: A 2 kg mass is attached to a spring (k = 100 N/m). It's displaced 0.5 m from equilibrium and released from rest. What's the total energy at the equilibrium position?

Solution:

• At equilibrium, all energy is kinetic, and potential energy is zero.
• Since total energy is conserved, it will be equal to the potential energy at the maximum displacement.
• $U = \frac{1}{2}kx^2 = \frac{1}{2}(100 \frac{N}{m})(0.5 m)^2 = 12.5 J$
• Total Energy = 12.5 J

#4. Final Exam Focus

#4.1 High Priority Topics

• Energy Conservation in SHM: Understand how potential and kinetic energy transform during SHM.
• Calculations: Be comfortable using the formulas for potential ($U = \frac{1}{2}kx^2$) and kinetic energy ($K = \frac{1}{2}mv^2$).
• Graphical Analysis: Interpret energy vs. time graphs.
• Conceptual Understanding: Know where potential and kinetic energy are maximized and minimized.

#4.2 Common Pitfalls

#5. Practice Questions

Practice Question

#Multiple Choice Questions

1. A mass-spring system oscillates with simple harmonic motion. At which point in the motion is the kinetic energy of the mass the greatest? (A) At the maximum displacement from equilibrium (B) At the equilibrium position (C) When the spring is compressed the most (D) When the mass is momentarily at rest

2. A spring with a spring constant k is compressed by a distance x and then released. If the spring is compressed by a distance of 2x, what is the ratio of the new potential energy to the original potential energy? (A) 1:1 (B) 1:2 (C) 1:4 (D) 4:1

3. A simple harmonic oscillator has a total energy E. If the amplitude of the oscillation is doubled, what is the new total energy of the oscillator? (A) E (B) 2E (C) 4E (D) E/2

#Free Response Question

A 0.5 kg block is attached to a horizontal spring with a spring constant of 200 N/m. The block is initially pulled 0.1 m from its equilibrium position and released from rest on a frictionless surface.

(a) Calculate the total energy of the block-spring system.

(b) Determine the maximum speed of the block as it oscillates.

(c) At what displacement from the equilibrium position will the block have half of its maximum kinetic energy?

(d) Sketch a graph of the potential energy, kinetic energy, and total energy of the system as a function of time over one period of oscillation. Clearly label each curve.

Scoring Breakdown:

(a) (2 points) * 1 point for using the correct potential energy formula: $U = \frac{1}{2}kx^2$ * 1 point for correct calculation: $U = \frac{1}{2}(200 N/m)(0.1 m)^2 = 1 J$

(b) (3 points) * 1 point for recognizing that total energy is conserved and equals the maximum kinetic energy. * 1 point for using the correct kinetic energy formula: $K = \frac{1}{2}mv^2$ * 1 point for correct calculation: $1 J = \frac{1}{2}(0.5 kg)v^2$, $v = 2 m/s$

(c) (3 points) * 1 point for recognizing that half of the maximum kinetic energy is equal to half of the total energy: 0.5 J * 1 point for setting the potential energy equal to half of the total energy: $0.5 = \frac{1}{2}kx^2$ * 1 point for correct calculation: $0.5 J = \frac{1}{2}(200 N/m)x^2$, $x = 0.071 m$

(d) (4 points) * 1 point for correct shape of potential energy curve (parabola with minimum at t = T/2) * 1 point for correct shape of kinetic energy curve (parabola with maximum at t = T/2) * 1 point for correct shape of total energy curve (horizontal line) * 1 point for labeling each curve correctly

Alright, you've got this! Remember to stay calm, focus on the core concepts, and trust your preparation. You're going to do great! 🎉