Oscillations are seen everywhere, from the music we hear to the springs we play with, and even with some topics we'll cover in electricity and magnetism 👀. In Unit 6, we will delve into the world of oscillations and learn about various aspects such as simple harmonic motion, springs, pendulums, and wave motion.

The big idea of this unit surrounds the following question: How does the presence of restoring forces predict and lead to harmonic motion?

Unit 6 will cover approximately 4-6% of the exam and should take around 4 to 7, 45-minute class periods to cover. The AP Classroom personal progress check has 20 multiple choice questions and 2 free response questions for you to practice on.

Key Vocabulary

• Simple harmonic motion: A type of periodic motion in which the restoring force is proportional to the displacement from equilibrium position and acts in the opposite direction to the displacement.
• Oscillation: The back-and-forth motion of an object about its equilibrium position.
• Amplitude: The maximum displacement of an oscillating object from its equilibrium position.
• Period: The time taken by an oscillating object to complete one cycle of motion.
• Frequency: The number of cycles per unit time of an oscillating object.
• Restoring force: The force that brings an oscillating object back to its equilibrium position.
• Spring constant: The constant that relates the force exerted by a spring to its displacement from its equilibrium position.
• Angular frequency: The rate at which an oscillating object completes one cycle of motion in radians per unit time.
• Phase angle: The initial angle of an oscillating object at the start of its motion.
• Resonance: A phenomenon in which an object is forced to vibrate at its natural frequency due to the application of an external force at the same frequency.

Some questions that can be explored in this unit are:

1. How does the period of oscillation of a mass-spring system vary with the mass of the object and the spring constant?
2. How does the period of oscillation of a simple pendulum vary with the length of the string and the acceleration due to gravity?
3. How does the energy of a mass-spring system vary during oscillation?
4. How does the amplitude of a mass-spring system vary with the initial displacement?
5. What is resonance and how does it occur in mass-spring systems and simple pendulums?

6.1 Simple Harmonic Motion, Springs, and Pendulums

Simple harmonic motion (SHM) is a type of periodic motion in which the restoring force is proportional to the displacement from equilibrium position and acts in the opposite direction to the displacement. This results in a sinusoidal motion that is characterized by a constant period and amplitude. The equation for SHM is given by x(t) = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase angle.

One of the most common examples of SHM is the motion of a mass attached to a spring. When the spring is stretched or compressed from its equilibrium position, it exerts a restoring force on the mass, which causes it to oscillate back and forth. The period of oscillation of the mass-spring system depends on the mass of the object and the spring constant.

Another example of SHM is the motion of a simple pendulum. A simple pendulum consists of a mass attached to a weightless, flexible string or rod that is suspended from a fixed point. When the mass is displaced from its equilibrium position, it experiences a restoring force due to gravity, which causes it to oscillate back and forth. The period of oscillation of the pendulum depends on the length of the string and the acceleration due to gravity.

Practice Problems

Here are some practice problems related to simple harmonic motion, springs, and pendulums:

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√(m/k), where m is the mass of the object and k is the spring constant. Plugging in the values, we get T = 2π√(0.2/20) = 0.628 s.
2. The period of oscillation of a simple pendulum is given by T = 2π√(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√(1/9.81) = 2.006 s.
3. The maximum velocity of a mass-spring system is given by vmax = Aω, where A is the amplitude and ω is the angular frequency (ω = 2π/T). Plugging in the values, we get vmax = (0.05 m) × (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. Therefore, the period of the pendulum on the moon would also be 2 s.
5. The maximum potential energy of a mass-spring system is given by Umax = 1/2kA^2, where k is the spring constant and A is the amplitude. Plugging in the values, we get Umax = (1/2) × 10 × (0.2)^2 = 0.2 J.