Chapter 2

Simple Harmonic Motion (SHM)

Learning Outcomes

Simple Harmonic Motion (amplitude, frequency, displacement, velocity, acceleration)

Conservation of energy in SHM Simple pendulum and spring Damped harmonic motion

Simple Harmonic Motion Position x vs. time t Definition of period T Definition of amplitude A

Simple Harmonic Motion Definitions of Terms

• Amplitude = A = the maximum displacement of the moving object from its equilibrium position.

• (unit = m)

• Period = T = the time it takes the object to complete one full cycle of motion.

• (unit = s)

• Frequency = f = the number of cycles or vibrations per unit of time.

• (unit = cycles/s = 1/s = Hz = hertz)

Simple Harmonic MotionMotion described by this equation:

is called simple harmonic motion (SHM). It is: periodic (repeats itself in time) oscillatory (takes place over a limited spatial

range)

tAx cosdisplacement (m)

amplitude (m)time (s)

angular frequency (rad/s)

SHM: Reference Circle Representation

A vector of magnitude

A rotates about the

origin with an angular

velocity

The x component of

the vector represents

the displacement.

t

A

A cos t

X

Y

SHM: Frequency

Since there are 2 radians in each trip (“cycle”) around the reference circle, the “cycle” frequency is related to the angular frequency by

SI units of “cycle” frequency, f:

cycles / s = Hertz (Hz)

2

or 2 ff

SHM: Velocity

We can calculate the

velocity from the

reference circle

representation: t

A

X

Y

tvT

- vT sin t

tAv

tAtv

Arv

T

T

sin

sinsin

SHM: Acceleration

tAa

taa

Ara

C

C

cos

cos

2

22

t

X

Y

taC

-aC cos t

General EquationsDisplacement from Equilibrium:

x(t) =

Velocity: v(t) = =

Acceleration: a(t) = =

Simple Harmonic Motion

dt

dx

dt

dv

A cos ( t + )

A sin ( t + )

A cos ( t + )

SHM Systems

The Period of a Mass on a SpringSince the force on a mass on a spring is proportional to the displacement, and also to the acceleration, we find that

Substituting the time dependencies of a and x gives

The Period of a Mass on a Spring

Therefore, the period is

Energy Conservation in Oscillatory Motion

In an ideal system with no nonconservative forces, the total mechanical energy is conserved. For a mass on a spring:

Since we know the position and velocity as functions of time, we can find the maximum kinetic and potential energies:

Energy Conservation in Oscillatory Motion

As a function of time,

So the total energy is constant; as the kinetic energy increases, the potential energy decreases, and vice versa.

The Pendulum

A simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass).

The angle it makes with the vertical varies with time as a sine or cosine.

Energy Conservation in Oscillatory Motion

This diagram shows how the energy transforms from potential to kinetic and back, while the total energy remains the same.

The Pendulum

A simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass).

The angle it makes with the vertical varies with time as a sine or cosine.

The Pendulum

Looking at the forces on the pendulum bob, we see that the restoring force is proportional to sin θ, whereas the restoring force for a spring is proportional to the displacement (which is θ in this case).

The Pendulum

However, for small angles, sin θ and θ are approximately equal.

The PendulumSubstituting θ for sin θ allows us to treat the pendulum in a mathematically identical way to the mass on a spring. Therefore, we find that the period of a pendulum depends only on the length of the string:

The Pendulum

A physical pendulum is a solid mass that oscillates around its center of mass, but cannot be modeled as a point mass suspended by a massless string. Examples:

The PendulumIn this case, it can be shown that the period depends on the moment of inertia:

Substituting the moment of inertia of a point mass a distance l from the axis of rotation gives, as expected,

Damped OscillationsIn most physical situations, there is a nonconservative force of some sort, which will tend to decrease the amplitude of the oscillation, and which is typically proportional to the speed:

This causes the amplitude to decrease exponentially with time:

Damped OscillationsThis exponential decrease is shown in the figure:

Damped OscillationsThe previous image shows a system that is underdamped – it goes through multiple oscillations before coming to rest. A critically damped system is one that relaxes back to the equilibrium position without oscillating and in minimum time; an overdamped system will also not oscillate but is damped so heavily that it takes longer to reach equilibrium.

Summary of Chapter 2• Period: time required for a motion to go through a complete cycle

• Frequency: number of oscillations per unit time

• Angular frequency:

• Simple harmonic motion occurs when the restoring force is proportional to the displacement from equilibrium.

Summary of Chapter 2• The amplitude is the maximum displacement from equilibrium.

• Position as a function of time:

• Velocity as a function of time:

Summary of Chapter 2

• Acceleration as a function of time:

• Period of a mass on a spring:

• Total energy in simple harmonic motion:

Summary of Chapter 2

• Potential energy as a function of time:

• Kinetic energy as a function of time:

• A simple pendulum with small amplitude exhibits simple harmonic motion

Summary of Chapter 2• Period of a simple pendulum:

• Period of a physical pendulum: