SHM in Pendula

Simple Harmonic

Motion

What is Simple Harmonic

Motion?

• Simple Harmonic Motion is motion that

repeats itself on an evenly timed basis.
– The Earth spins around every 24 hours.
– The Earth revolves around the Sun every

365.249 days

– A pendulum swings back and forth

What causes Simple Harmonic

Motion?

• Simple Harmonic Motion

(SHM) is caused by a
restoring force.

• A restoring force is a force

that pushes an object back
to its resting state –
equilibrium.

Key Words to Know

• Amplitude – how far away from

equilibrium the pendulum is pulled

• Cycle – the complete motion; all the way

until it begins again.
– The same position and direction!!

• Period – how long (TIME!!) it takes for

the pendulum to complete one cycle.

• Frequency – how many times the

pendulum swings back and forth in one
second

Why does a pendulum swing back

and forth?

• Gravity! Gravity is the restoring force.

– The pendulum wants to hang straight down.

When we pull it to the side, we are exerting
an applied force.

– When we let go, gravity pulls it back down.
– Because of inertia, the pendulum keeps

swinging till it gets to the other side.

– When it goes as high as it can on the other

side (the same potential energy), gravity
pulls it back.

A simple pendulum

A simple pendulum undergoes SHM (simple

harmonic motion) iff all of the following
conditions are met:

1)

The string’s mass is negligible compared to
the bob.

2)

The string doesn’t stretch.

3)

Friction between the string and the support
is negligible.

4)

The initial angle is very small (less than 15o)

SHM of a Pendulum

•

Implications

As the length of the pendulum increases,

the period increases. Does this make
sense?

As the length increases, the bob has to
travel farther in order to complete the
cycle. Therefore, it should take longer!

As for the frequency, since the pendulum has

longer to go, it can complete fewer cycles,
and so the frequency decreases.

Example

• What is the period on Earth of a

pendulum that is 2m long?

• What is the period of that same

pendulum on the moon, where gravity is
1.6m/s2?

Solution

• What is the period on Earth of a

pendulum that is 2m long?

Example

What is the period of that same
pendulum on the moon, where gravity is
1.6m/s²?

Multiple Choice

If the length of a pendulum increases, its period....

• increases

• decreases

• stays the same

Multiple Choice

The restoring force on a pendulum is

• increasing

• gravity

• electrical

• friction

Multiple Choice

If a pendulum is moved to a planet and its period increases, the gravity on that planet is ___________ the gravity on Earth.

• the same as

• less than

• greater than

• can't tell; period doesn't depend on gravity

Multiple Select

Which of the following are most likely to result in simple harmonic motion? Select two answers.

• A hole is drilled through one end of a meter stick, which is hung vertically from a frictionless axle. The bottom of the meter stick is displaced 12 degrees and released.

• A hole is drilled through one end of a meter stick, which is hung vertically from a rough axle. The bottom of the meter stick is displaced 12 degrees and released. Every time the meter stick swings back and forth the axle squeaks.

• A block is hung vertically from a linear spring. The opposite end of the spring is attached to a stationary point. The entire apparatus is placed in deep space. The block is displaced 4 cm from equilibrium and released.

• A block is placed on a frictionless surface and attached to a non-linear spring. The opposite end of the spring is attached to a wall. The block is displaced 2 cm from equilibrium and released.

Multiple Choice

Student X attaches an object of mass M to the end of a string of length L so that a simple pendulum is constructed. Student Y attaches an object of mass M to a string of length 4L to construct a second pendulum.

Which of the following claims correctly compares the periods of both pendula?

• The period of Student X’s pendulum is the same as the period of Student Y’s pendulum.

• The period of Student X’s pendulum is four times the period of Student Y’s pendulum.

• The period of Student X’s pendulum is twice the period of Student Y’s pendulum.

• The period of Student X’s pendulum is half the period of Student Y’s pendulum.

Multiple Choice

The gravitational field strength near Jupiter’s surface is nearly 2.53 times greater than the gravitational field strength near Earth's surface. Which of the following claims is correct about the period of a pendulum if it oscillates near Jupiter’s surface and near Earth’s surface?

• The period of the pendulum will be greater on Jupiter.

• The period of the pendulum will be greater on Earth.

• The period of the pendulum is the same on both planets.

• A comparison of the period of the pendulum for both planets cannot be made without more information about the length of the pendulum.

Multiple Choice

The angle between the string of a pendulum at its equilibrium position and at its maximum displacement is the pendulum’s

• period.

• frequency.

• vibration.

• amplitude.

Multiple Choice

For a system in simple harmonic motion, which of the following is the time required to complete a cycle of motion?

• amplitude

• period

• frequency

• revolution

Multiple Choice

For a system in simple harmonic motion, which of the following is the number of cycles or vibrations per unit of time?

• amplitude

• period

• frequency

• revolution

Multiple Choice

A certain pendulum with a 1.00 kg bob has a period of 3.50 s. What will happen to the period of the pendulum if the 1.00 kg bob is replaced by a bob with a mass of 2.00 kg?

• Increase

• Decrease

• Stays the same

• Need more information

Multiple Choice

Suppose that a pendulum has a period of 4.0 s at Earth’s surface. If the pendulum is taken to the moon, where the acceleration due to gravity is much less than on Earth, will the pendulum’s period increase, decrease, or stay the same? 

• Increase

• Decrease

• Stays the same

• Need more information

Fill in the Blank

A pendulum has a length of 0.84m. What is its period in seconds? Round your answer to 2 decimal places.