Simple Harmonic Motion Elasticity

- Chapter 10

Elastic Potential Energy

- What is it?
- Energy that is in materials as a result of

their . - Where is it found?

Law

- A spring can be or with a .
- The by which a spring is compressed or

stretched is to the magnitude of the

( ). - Hookes Law
- Felastic
- Where
- spring constant of spring ( )
- displacement

Hookes Law

- What is the graphical relationship between the

elastic spring force and displacement? - Felastic -kx

Hookes Law

- A force acting on a spring, whether stretching or

compressing, is always . - Since the spring would prefer to be in a

relaxed position, a negative force will

exist whenever it is deformed. - The force will always attempt to bring

the spring and any object attached to it back to

the position. - Hence, the restoring force is always .

Example 1

- A 0.55 kg mass is attached to a vertical spring.

If the spring is stretched 2.0 cm from its

original position, what is the spring constant? - Known
- m
- x
- g
- Equations
- Fnet (1)
- (2)
- (3)
- Substituting 2 and 3 into 1 yields
- k
- k
- k

Elastic in a Spring

- The exerted to put a spring in tension or

compression can be used to do . Hence the

spring will have Elastic . - Analogous to kinetic energy

Example 2

- What is the maximum value of elastic potential

energy of the system when the spring is allowed

to oscillate from its relaxed position with no

weight on it?

- A 0.55 kg mass is attached to a vertical spring

with a spring constant of 270 N/m. If the spring

is stretched 4.0 cm from its original position,

what is the Elastic Potential Energy? - Known
- m 0.55 kg
- x -4.0 cm
- k 270 N/m
- g 9.81 m/s2
- Equations
- PEelastic
- PEelastic
- PEelastic

Elastic Potential Energy

- What is area under the curve?

A A A A Which you should see equals the

Simple Harmonic Motion Springs

- Simple Harmonic Motion
- An around an will occur when an

object is from its equilibrium position and

. - For a spring, the restoring force F -kx.
- The spring is at equilibrium
- when it is at its relaxed length.
- ( )
- Otherwise, when in tension or
- compression, a restoring
- force exist.

Simple Harmonic Motion Springs

- At displacement ( )
- The Elastic Potential Energy will be at a
- The force will be at a .
- The acceleration will be at a .
- At (x )
- The Elastic Potential Energy will be
- Velocity will be at a .
- Kinetic Energy will be at a
- The acceleration will be , as will the

force.

10.3 Energy and Simple Harmonic Motion

Example 3 Changing the Mass of a Simple Harmonic

Oscilator

A 0.20-kg ball is attached to a vertical spring.

The spring constant is 28 N/m. When released

from rest, how far does the ball fall before

being brought to a momentary stop by the spring?

10.3 Energy and Simple Harmonic Motion

Simple Harmonic Motion of Springs

- Oscillating systems such as that of a spring

follow a pattern. - Harmonic Motion of Springs 1
- Harmonic Motion of Springs (Concept Simulator)

Frequency of Oscillation

- For a spring oscillating system, the frequency

and period of oscillation can be represented by

the following equations - Therefore, if the of the spring and the

are known, we can find the and

at which the spring will oscillate. - k and mass equals frequency of

oscillation (A spring).

Harmonic Motion The Simple Pendulum

- Simple Pendulum Consists of a massive object

called a suspended by a string. - Like a spring, pendulums go through
- as follows.
- Where
- Note
- This formula is true for only of .
- The period of a pendulum is of its mass.

Conservation of ME The Pendulum

- In a pendulum, is converted into

and vise-versa in a continuous repeating pattern. - PE mgh
- KE ½ mv2
- MET PE KE
- MET
- Note
- kinetic energy is achieved at the point of

the pendulum swing. - The potential energy is achieved at the

of the swing. - When is , , and when

is , .

Key Ideas

- Elastic Potential Energy is the in a spring

or other elastic material. - Hookes Law The of a spring from its

is the applied. - The of a vs. is equal to the

. - The under a vs. is equal to the

done to compress or stretch a spring.

Key Ideas

- Springs and pendulums will go through oscillatory

motion when from an position. - The of of a simple pendulum is of

its of displacement (small angles) and . - Conservation of energy Energy can be converted

from one form to another, but it is

.