# Activity: SHM, energy and phase

#### Aims

Investigate other oscillations.

Explore the changing energy in an oscillation.

Sketch graphs of KE and PE vs time and position.

Introduce phase difference.

#### Simple harmonic motion

Simple harmonic motion is defined as motion where the acceleration is directly proportional to the displacement from a fixed point and always acts towards that point.

a = -ω^{2}x

#### Is it SHM?

Not all oscillatory motion is SHM but if it is you can use the equations for SHM to find the frequency of oscillations. Let's look at some examples.

**Mass on a spring.**

When a mass hanging on a spring is displaced it oscillates, you can try this using the PhET masses and springs simulation.

Notice that with friction the amplitude reduces with time, this is *damped harmonic motion*. You will learn about this if you do the HL Engineering option. Reducing friction to zero removes damping.

What happens to the frequency if you

- increase the mass
- make the spring more soft

Consider the forces at the lowest point

- Write an equation for the forces at the equilibrium position.
- Apply Hooke's law to write and expression for F
_{1}in terms of the spring constant k. - In which direction is the acceleration at the bottom of the oscillation?
- Apply Newton's law to write an expression for the acceleration.
- Apply Hooke's law to write an expression for F
_{2}. - This equation shows that the motion is SHM, why?
- Show that

- By taking measurements from the simulation show that this equation applies. Of course it does so see if it applies to a real spring as well.

**Floating block**

Using the buoyancy simulation you can make a floating block of wood oscillate.

This time you can't remove the friction so the motion is damped but if it wasn't damped would this be SHM?

Fill in the missing words to explain why the floating block executes SHM

word list: acceleration Archimedes first displacement SHM volume buoyant weight

According to principle; when the block is floating the of water displaced equals the weight of the block. When the block is pushed down the force increases. The forces are now unbalanced so according to Newton's law the block will accelerate upwards. The will be proportional to the unbalanced force which is proportional to the extra volume displaced. The block has a regular shape so the volume displaced is proportional to the downwards. So is proportional to displacement which means this is .

#### Energy changes

Consider the simple pendulum again.

Before the pendulum was set in motion the bob was in position O. It was the pushed to A (by me) and released.

- Was work done on the pendulum pushing it to A?
- What form of energy did the pendulum gain?
- What form of energy does the pendulum have as it swings past O?

As the pendulum swings there is a continual exchange of energy from PE to KE and back to PE. Let's first look at the KE. The total energy remains constant

Total Energy = PE + KE

__Variation of KE with time__

- Write the equation for KE?
- Write the equation representing how the velocity of a body with SHM changes with time. Take the starting point to be as the bob passes the centre travelling from A to B.
- Substitute for v in the KE equation to show that

Note that it is cos because we start in the middle where KE is max.

- What is the maximum value for KE?

__Variation of PE with time__

- What is the maximum value of PE?

Since energy is conserved the PE at any time will be PE_{max} - KE

- Write an equation for the PE at time t and show that

__Graphical representation__

Observe the changes in PE and KE as you vary mass, amplitude and frequency.

Exercise 10 page 160

#### Phase

If two objects oscillate in time with each other they are said to be in phase if not then they are out of phase.

You can see that the displacement time graphs are shifted by π radians, this is the *phase difference* or *phase angle*.

The equation for the displacement of the blue body is y = x_{o}sinωt

- What is the equation for the red body?

In the next simulation you can change the phase angle.

See what happens when the phase difference of these pendulums changes.