Harmonic Motion

Simple Harmonic Motion

​Simple Harmonic Motion

When an object is displaced, it may be subject to a restoring force, resulting in a periodic oscillating motion.
If the displacement is directly proportional to the linear restoring force, the object undergoes Simple Harmonic Motion.
Ex: pendulum on a grandfather clock

Oscillator vs Cycle

A system in simple harmonic motion is called an oscillator.
Examples are a pendulum or a mass hanging from a spring.

A cycle is a unit of motion that repeats over and over.
An orbit and a rotation are both cycles because they are repeating motions.

Categorize

Options (6)

orbit

pendulum

spring

rotation

El nino

rubber banc

Organize these options into the right categories

Oscillators

Cycles

​Oscillation

Oscillation occurs whenever a system has restoring forces that give it stability and act to return it to the equilibrium position.
Stability is a characteristic of harmonic motion.

Stability

The concept of stability is critical to flight. Wind gusts often pitch a plane up or down.
If the fins are in the front, any slight pitch of the plane results in a force that causes the plane to rotate more. This is called positive feedback.
When the fins are in the back, air flowing over the tail creates forces that push the plane back toward level flight. These restoring forces create negative feedback.

Period vs Frequency

The time for one cycle is called the period.
The frequency of an oscillator tells you the number of cycles it completes each second.
The unit of frequency is hertz. (Hz)
The hertz corresponds to one cycle per second.
f=1/T

Math Response

A sprinter's heart beats once every 0.33s. What is the frequency of her heart beat?

Type answer here

Correct Answer: 3.0

Deg^°

Rad

Equilibrium

Most oscillators have a resting state or equilibrium position.
At equilibrium the net force is zero.
A system in equilibrium remains in equilibrium until some outside force disturbs it.
Any force that disturbs the system's equilibrium adds energy and the added energy is what causes the oscillations.

Amplitude

The amplitude describes how far an oscillator moves away from equilibrium during each cycle.
Amplitude is measure in units that match the oscillation.
The key idea is that amplitude always describes the maximum displacement from equilibrium. This is usually half the distance to the lowest point.

Labelling

Label the oscillator.

Drag labels to their correct position on the image

maximum amplitude 2

equilibrium

maximum amplitude 1

length of string

bob

Harmonic motion involves energy that oscillates among different forms.

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​Energy in an oscillator

Damping

Friction converts kinetic energy into heat and wear.
As the energy of an oscillator decreases from friction, the amplitude decreases.
The frictional decrease in amplitude is called damping.
Over time, damping reduces the speed of an oscillating mass on a spring until it gradually comes to a stop.

Multiple Choice

What is the period of a pendulum that is 4 meters long? (Assume $$g = 9.8 \, \text{m/s}^2$$ )

• 2.01 s

• 4.02 s

• 6.28 s

• 8.04 s

Multiple Choice

A mass attached to a spring oscillates with a frequency of 2 Hz. What is its period?

• 0.25 s

• 0.5 s

• 1 s

• 2 s

Multiple Choice

What is the amplitude of a simple harmonic oscillator whose maximum displacement from the equilibrium position is 5 cm?

• 2.5 cm

• 5 cm

• 10 cm

• 20 cm

Multiple Choice

What is the equation for the period of a mass-spring system?

• $$T = 2\pi\sqrt{\frac{m}{k}}$$

• $$T = 2\pi\sqrt{\frac{k}{m}}$$

• $$T = \frac{1}{2\pi}\sqrt{\frac{m}{k}}$$

• $$T = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$$

Multiple Choice

In damped harmonic motion, what happens to the amplitude over time?

• It increases.

• It remains constant.

• It decreases.

• It oscillates.

Pendulum

A pendulum has a mass suspended below a pivot point by a rope or chain, allowing the mass to oscillate back and forth.
The equilibrium point is directly below the pivot, where the mass hands at rest.
The period is the time it takes to complete one full back and forth swing.

Pendulum and SHM

The weight mg acts vertically down while tension T acts along the string.
When the string makes an angle Θ_max to the right, the forces along the direction of the string cancel, leaving a net force F_net that points down and to the left, back towards equilibrium. When the string makes the same angle to the left, the net force points down and to the right. The net force pushes the bob back toward equilirbium.

Multiple Choice

What is the frequency of a pendulum that completes 20 oscillations in 40 seconds?

• 0.5 Hz

• 1 Hz

• 2 Hz

• 4 Hz

Multiple Choice

Which of the following is true for a pendulum at its highest point in the swing?

• Kinetic energy is maximum, and potential energy is minimum.

• Kinetic energy is minimum, and potential energy is maximum.

• Both kinetic and potential energy are equal.

• Neither kinetic nor potential energy is at its maximum.

Multiple Choice

If the mass of a pendulum is doubled, what happens to its period?

• It doubles.

• It halves.

• It remains the same.

• It quadruples.

Multiple Choice

What is the maximum velocity of a simple harmonic oscillator with an amplitude of 0.1 m and a frequency of 5 Hz?

• $$3.14 \, \text{m/s}$$

• $$1.57 \, \text{m/s}$$

• $$0.5 \, \text{m/s}$$

• $$0.31 \, \text{m/s}$$

Natural Frequency

The natural frequency is useful because many inventions are designed to work at a specific frequency.
The process of tuning is really a process of adjusting the natural frequency of a vibrating string oscillator.

Multiple Choice

In the context of damped harmonic motion, what role does the damping coefficient play?

• It increases the period of oscillation.

• It decreases the period of oscillation.

• It has no effect on the period of oscillation.

• It determines the rate at which the amplitude decreases.

Multiple Choice

What is the relationship between the period of a pendulum and its length?

• The period is directly proportional to the square root of its length.

• The period is inversely proportional to the square root of its length.

• The period is directly proportional to the length.

• The period is inversely proportional to the length.

Multiple Choice

Which of the following is a characteristic of periodic motion?

• The motion is always linear.

• The motion repeats after equal intervals of time.

• The motion is always accelerated.

• The motion never repeats.

How to calculate period of a pendulum

The restoring force on the pendulum bob is a component of the gravitational force.
F_restoring=mgsinΘ=mg(x/L)
a_max=g(x/L)
a=gx/2L
t=√(2x/(gx/2L)

t_period =2π√L/g

Multiple Choice

What does the term "amplitude" refer to in simple harmonic motion?

• The maximum speed of the oscillator.

• The maximum acceleration of the oscillator.

• The maximum displacement from the equilibrium position.

• The total energy of the oscillator.

Multiple Choice

If the length of a pendulum is quadrupled, by what factor does the period change?

• It doubles.

• It halves.

• It remains the same.

• It quadruples.

Natural frequency of a Spring

At maximum amplitude the spring is extended a distance x. At this point the velocity of the mass is zero, and the energy of the system is entirely elastic potential energy.
Max displacement: E=1/2kx²
v_max=x√k/m
T=2π√m/k
f=1/(2π)√k/m

Multiple Choice

In a simple harmonic oscillator, when is the acceleration zero?

• At the equilibrium position.

• At the maximum displacement.

• At the maximum velocity.

• Acceleration is never zero in simple harmonic motion.

Multiple Choice

What is the relationship between the frequency $$f$$ and the period $$T$$ of a simple harmonic oscillator?

• $$f = \frac{1}{T}$$

• $$f = T$$

• $$f = 2\pi T$$

• $$f = \frac{T}{2\pi}$$

Multiple Choice

Which of the following scenarios describes a damped harmonic motion?

• A pendulum swinging with no air resistance.

• A mass oscillating on a spring without friction.

• A pendulum swinging in air, gradually coming to a stop.

• A mass oscillating on a spring in a vacuum.

Periodic Forces

A periodic force is a force that repeats in cycles, like the repeated push, wait, push, wait, push that you use to get a swing going.
When the frequency of the force matches a system's natural frequency, even a small force can produce a surprisingly large oscillation.

​Resonance

A resonant system accumulates energy with every cycle of applied force.
A system in resonance is a very efficient energy accumulator.

Quantities that are a maximum at (a) and (c)

• Displacement

• Force ( F = kx)

• Acceleration ( a = F/m)

• Elastic Potential Energy ( U_s = (1/2)kx²)

Quantities that are zero at (a) and (c)

• Velocity

• Kinetic Energy ( K = (1/2)mv²)

• Momentum ( p = mv)

Quantities that are a zero at (b)

• Displacement

• Force

• Acceleration

• Elastic Potential Energy

Quantities that are a maximum at (b)

• Velocity

• Kinetic Energy

• Momentum

A pendulum closely follows SHM

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• l = length of pendulum

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• k = spring constant

Multiple Choice

A simple harmonic oscillator takes 4.8 s to undergo five complete vibrations. What is the frequency? 

• 1 Hz

• 0.96 Hz

• 1 s

• 0.96 s

Multiple Choice

What is the period on Earth of a pendulum with a length of 2.4 m?

• 3.1 s

• 1.5 s

• 0.77 s

• 1.2 sec

Multiple Select

What variables affect the period of a pendulum? Check all that apply.

• the mass of the pendulum bob

• the length of the string

• the gravitational acceleration

• the type of string used

Multiple Select

What variables affect the period of a mass on a spring? Check all that apply.

• the mass of the object on the spring

• the mass of the spring

• the spring constant

• how far the spring is stretched

Multiple Choice

The velocity of a spring-block system vibrating along the y-axis changes with time according to the equation: 𝑣(𝑡) = 2𝜋 cos(𝜋𝑡 + 𝜋). All quantities in the equation are expressed in S.I units. Consider that the object is initially at the equilibrium position. What is the maximum acceleration of the object?

• 𝜋

• 2𝜋

• 2𝜋²

• 𝜋/2

Multiple Choice

The velocity of a spring-block system vibrating along the y-axis changes with time according to the equation: 𝑣(𝑡) = 2𝜋 cos(𝜋𝑡 + 𝜋). All quantities in the equation are expressed in S.I units. Consider that the object is initially at the equilibrium position. What is the value of the velocity (in m/s) at the instant when the acceleration is zero?

• 2𝜋

• 2𝜋²

• 𝜋

• 𝜋/2

Multiple Choice

A force of 16 N is required to stretch a spring a distance of 40 cm from its rest position. What force (in Newtons) is required to stretch the same spring 60cm?

• 24 N

• 16 N

• 32 N

• 4 N