Vibrations and Waves

Sound Section 1

What do you think?

• What is sound?• What do all of the sounds that you hear have in

common?• How do they differ?

• Can sounds travel through solids? Liquids? Gases?• Is one type of material better for transmitting sound

waves?

• When race cars or emergency vehicles pass you, the sound changes. In what way, and why?

Simple Harmonic Motion

• Vibrations about an equilibrium position which a restoring force is proportional to the displacement from equilibrium

• What does this mean?– Periodic motion is back and forth

over the same path.

Simple Harmonic Motion

• Equilibrium position, velocity reaches a maximum.

• Let’s look at an example.– Spring

Simple Harmonic Motion

• Equilibrium position is when the spring is unstretched

• When a spring has a mass attached at the end

Simple Harmonic Motion

• If the spring is stretched and released, the spring exerts a force on the mass towards the equilibrium position.

Simple Harmonic Motion

• The spring force decrease, as the spring reaches the equilibrium position. Eventually reaching zero.

Simple Harmonic Motion

• The mass’s acceleration also reaches zero at equilibrium.

Simple Harmonic Motion

• However, the velocity of the mass increases as it approaches equilibrium. Reaching a maximum velocity at equilibrium.

Simple Harmonic Motion

• At maximum displacement, spring force and acceleration reach a maximum.

Simple Harmonic Motion

• This can occur in the opposite direction (compression).

Simple Harmonic Motion

• Spring’s compression is equal to the distance the spring was stretched.

Simple Harmonic Motion

• Ideally, the mass-spring system would oscillate indefinitely.

Simple Harmonic Motion

• But friction retards, the motion and eventually the mass-spring will come to a rest. (Damping)

Simple Harmonic Motion

• Restoring force- spring pushes and pulls a mass back to equilibrium.

Simple Harmonic Motion

• Restoring force- directly proportional to the displacement of the mass.

Hooke’s Law

• F elastic = -kx

Spring force = - (spring constant)(displacement)

Negative sign signifies direction of the spring force opposite the direction of the mass displacement.

Hooke’s Law

• K = spring constant (always positive)

• Measure of the stiffness of the spring

• Greater value = stiffer the spring

• SI units of K is N/m

Simple Harmonic Motion

• All systems we will consider mechanical energy is conserved.

• PE initial = KE final

Example 1

If a mass of 0.55kg attached to a vertical spring stretches the spring 2.0cm from its equilibrium position, what is the spring constant?

K= 270 N/m

Example 2

If a mass of 0.55kg attached to a vertical spring stretches the spring 36.0cm from its equilibrium position, what is the spring constant?

K= 15 N/m

Example 3

Is the second spring stiffer or less stiff?

Less Stiff

Example 3

A slingshot consists of a light leather cup attached between two rubber bands.

If it takes a force of 32 N to stretch the bands 1.2 cm, what is the spring constant of the rubber band?

2666.67 N/m

Example 4

A slingshot consists of a light leather cup attached between two rubber bands.

How much force is required to pull the cup of this slingshot 3.0 cm from its equilibrium position?

81 N

Example 5

A load of 45N attached to a spring that is hanging vertically stretches the spring 0.14 m. What is the spring constant?

321.43 N/m

Simple Pendulum

• Pendulum consist of a mass (bob) attached to a fixed string

• Disregard friction and the mass of the string

Simple Pendulum

• Is a simple pendulum, simple harmonic motion?– What is the restoring force?– Is the restoring force proportional

to the displacement?– If so, then YES!

Simple Pendulum

• Is a simple pendulum, simple harmonic motion?– What is the restoring force?– Is the restoring force proportional

to the displacement?– If so, then YES!

Simple Pendulum

• Does the force acting on the bob act as the restoring force?– Force exerted by the string acts

along the y-axis– Any point other than equilibrium,

bob’s weight can be resolved into two components

Simple Pendulum

• Does the force acting on the bob act as the restoring force?– Force exerted by the string and the

y component of the bob’s weight are perpendicular to the motion

– X component of the bob’s weight is the net force acting on the bob in the direction of motion.

Simple Pendulum

• Does the force acting on the bob act as the restoring force?– Therefore, the X component pushes

or pulls the bob towards equilibrium

– It is a restoring force.

Simple Pendulum

• Restoring force of a SP is SHM• Small angles, the pendulum is

SHM• Less than 15°

Example 6

Are the following examples of simple harmonic motion?

A child swinging on a playground swing at a small angle.

Yes

Example 7

Are the following examples of simple harmonic motion?

An oscillating clock pendulum.

Yes

Example 8

Are the following examples of simple harmonic motion?

A rotating record on a turntable

No

Example 9

A pinball machine uses a spring that is compressed 4.0 cm to launch a ball. If the spring constant is 13 N/m, what is the force on the ball at the moment the spring is released?

0.52 N

Example 10

How does the restoring force acting on a pendulum bob change as the bob swings towards the equilibrium position?

Force decreases

Example 11

How does the bob’s acceleration change as the bob swings towards the equilibrium position?

Acceleration decreases

Example 12

How does the bob’s velocity change as the bob swings towards the equilibrium position?

Velocity increases