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# Oscillations

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by

## Liam Fishwick

on 26 August 2015

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#### Transcript of Oscillations

What is simple harmonic motion?
To investigate the relationship between circular motion and SHM
To be able to calculate the displacement at any particular moment under different starting conditions
To be able to calculate the velocity at any particular moment.

Equations of SHM
Objectives:
Be able to arrive at the equations for SHM

Calculate the magnitude of acceleration, or velocity at any time for an object in SHM
Oscillations
Specification points
119. Recall that the condition for simple harmonic motion is F = -kx, and hence identify situations in which simple harmonic motion will occur.
120. Recognise and apply the equations

as applied to a simple harmonic oscillator.
121. Obtain a displacement-time graph for an oscillating object and recognise that the gradient at a point gives the velocity at that point.
122. Recall that the total energy of an undamped simple harmonic system remains constant and recognise and use expressions for total energy of an oscillator.
123. Distinguish between free, forced, and damped oscillaions.
124. Investigate and recall how the amplitude of a forced oscillation changes at and around the natural frequency of a system and describe, qualitatively, how damping affects resonance.

Objectives:

Be able to derive the expression for the time period of a mass-spring system (C)

Investigate the relationship between mass and time period. (C)

Apply this knowledge to exam questions. (A)

Examples of SHM
There are many examples of simple harmonic motion, (SHM) in every day life & this topic connects well with earlier work on waves & circular motion.
Examples :
Child on a swing
A mass on a spring
A simple pendulum
A boat rocking side to side on water
A ball rolling from side to side in a dish
Child on a swing
Which of the examples are SHM?
Try the different experiments.
Set each in motion, which are SHM?
What happens if you change the amplitude?
The child oscillates about an equilibrium position, it moves repeatedly through this position in one direction then the other.
Phase difference
Imagine a set of two identical twins on two identical swings. (Their oscillation will be identical...)

Consider that one twin starts swinging before the other, they will oscillate "out of phase" by a difference between each other.

The phase difference will remain the same and can be described as a fraction of the whole period.
amplitude
frequency
period
What would their displacement-time graph look like?
Problems
Think of two further everyday examples of SHM.
1. Describe the motion of a bungee jumper after he leaves the platform:
Initially describe it for an ideal world, (in a vacuum with a perfectly elastic rope)
Secondly describe what is observed in the real world and account for the differences between ideal and real world scenarios

2. A mass on a spring is displaced and takes 9.6s to complete 20 complete oscillations. Calculate:
The time period & frequency of oscillation
3. Consider two identical pendulums, (X & Y) which complete 20 complete cycles in 16s. What is the phase difference if:
X passes through equilibrium 0.2s later than Y in the same direction
X & Y both reach maximum displacement at the same time but in opposite directions.

How does this link with Newton's laws?

When is the velocity of an object at a maximum?

When is the velocity of an object at a minimum?

Is velocity constant? Or does it depend on something else?
Complete the table for a swing (pendulum) with relative size and direction
The restoring force is a necessary condition for SHM.

Acts towards equilibrium
We can assign a constant to this.
Graphs of motion in SHM
Above is the displacement-time graph for a pendulum (note how it is a sine curve).

Sketch this, and below, work out what the velocity-time graph and acceleration-time graph would look like.
Graphs of motion in SHM
Velocity is gradient of displacement-time graph.
acceleration is the gradient of velocity-time graph
max velocity when displacement is zero. min velocity when displacement is max.
max acceleration when velocity is zero, always towards the centre. min acceleration when velocity is max.
The maths behind it
Conclusions
The acceleration is always in the opposite direction of to the displacement

SHM is defined as an oscillating motion where the acceleration is :

1. Proportional to displacement
2. Always in the opposite direction to displacement
Pendulum is an example of SHM, can be linked to circular motion: acceleration given by:
Problems
1. A mass on a spring oscillates in SHM with amplitude 25mm & a period of 2s. If it passes the equilibrium in the upwards direction at time t=0 what is the displacement of the object:
¼ cycle later, ½ cycle later, ¾ cycle later & 1 cycle later?

Calculate :

The frequency
Acceleration when displacement is +25mm, 0 & -25mm

2. A simple pendulum oscillates in SHM with an amplitude of 32mm. It takes 20s to complete 10 complete oscillations. Calculate:
The frequency
The initial acceleration

If the pendulum is released at time t=0, state the displacement and calculate the acceleration when t = 1.0s and t = 1.5s
Extension: Complete questions 1-10 on page 102 Advanced physics for you
http://www.physics.uoguelph.ca/tutorials/shm/phase0.html
Circular motion
Object in circular motion, at any time x and y co-ordinates are:
x
y
Starting point determines equation.
zero displacement - sin
Max displacement - cos
Velocity of object
Where
v
is the velocity,
f
is the frequency,
A
is the maximum displacement and
x
is the displacement

The velocity is considered positive if moving away from the equilibrium point and negative if moving towards it

Note this will be a maximum when x=0
Acceleration
In a similar way the maximum value for acceleration given by the acceleration equation .
Questions
1. A spring oscillates in SHM with a period of 3s and an amplitude of 58mm. Calculate:
The frequency
The maximum acceleration

2. The displacement of an object oscillating in SHM changes with time and is described by X (mm) = 12 cos 10t where t is the time in seconds after the object’s displacement was at its maximum value. Determine:
The amplitude
The time period
The displacement at t = 0.1s
3. An object on a spring oscillating in SHM has a time period of 0.48s and a maximum acceleration of 9.8m s^-2. Calculate:
Frequency
Amplitude
4. An object oscillates in SHM with an amplitude of 12mm and a period of 0.27s. Calculate
The frequency
Its displacement and direction of motion 0.1s, and 0.2s after the displacement was +12mm

The mass-spring system
How do you weigh yourself in space?
What does this tell us about the time period of the mass-spring system and gravitational fields?
Investigate the relationship
You have a hacksaw blade oscillator. You have a set of masses and a timer.

Task: Work out the mass of the unknown "masstronaut"

Follow the instructions on the sheet.
Task 1: Starting from Hooke's law, can you derive an expression for the time-period of a mass spring system?
Task 1: Starting from Hooke's law, can you derive an expression for the time-period of a mass spring system?
Hooke's law:
the change in length produced by a force on a wire or spring is directly proportional to the force applied.
Can you link this to the restoring force in SHM?
What is the condition for SHM?
How does this link acceleration and Newton's 2nd law?
The time period for a mass spring system
Interatomic bonds
Intertomic bonds can be modelled as mass-spring systems. Use NaCl as a simple example.
Typical interatomic forces:
Mass of Na (in NaCl)
Calculate the natural frequency of the atoms.
IR region of EM spectrum, what does this tell us?
Practice questions
Have a go at the exam questions.
The simple pendulum and energy
122. Recall that the total energy of an undamped oscillator remains constant (C)

Understand which factors affect the period of a pendulum (C)

Be able to use this to calculate g experimentally (B)
Factors affecting the period of a pendulum
With the pendulum hanging from the ceiling....

How would you measure the restoring force?

How can we measure the period?

What can we change and see if it changes the period?
The period of a pendulum
These were what we can change
1. Mass of the bob
2. Length of the string
3. Initial displacement (or amplitude)

Which affect the period?
L
s
Consider a simple pendulum with a bob of mass m suspended by a thread of length L.

Which has a displacement s from the equilibrium position giving an angle to the vertical
Period of a pendulum
The weight of the bob has components
perpendicular:
parallel to the motion:

The restoring force F causing the SHM will be in the opposite direction:
Use a small angle approximation
Period of a pendulum
Period of a pendulum
From the equation we can see that:

The period is independent of mass
The period is independent of initial displacement (amplitude)
The period is dependent only upon the length of string and g.

This can be used to establish a value for g experimentally

Do it! determine g by pendulum.....
Improvements to the experiment.
What improvements can you come up with?

Why are pendulums in clocks not spherical?
Problems.
Calculate the time period of a simple pendulum with lengths
1.0m
0.25m
Take g to be 9.81 m s^-2
Now calculate the period for the 1.0m pendulum on the moon where gravity is around 1/6th of that on Earth (g=1.6 m s^-2)

Move onto TAP 304-3 when complete.
Energy of SHM
If we consider an ideal pendulum....
Which form of energy will it have when it is displaced from the equilibrium position at one of the two extremes of oscillation
Which form of energy will it have when it is moving through the equilibrium position
Which form(s) of energy will it have during the rest of the cycle?
How does the total energy change during the whole cycle?
The energy graph
what would the answers to these be for a horizontal spring, and then a vertical spring?
Total energy remains a fixed constant (assuming no losses)
losses - this is known as damping
Complete questions TAP 305-5
Damping and forced oscillations
123. Distinguish between free, forced, and damped oscillaions.

To be able to describe what damping is an its effect on an oscillator (C)

To apply this knowledge to exam questions (A)
Questions.....
What equipment do cars or bikes tend to have to iron out the bumps in the road?
If the car suspension consisted only of springs, what would happen to the motion of the car for sometime after hitting a bump in the road?
How would a pendulum continue to swing in an “ideal physics world” with no friction & in a vacuum
What do we observe in the “real-world”
Most vehicles use springs or similar to iron out the lumps and bumps

With only springs the vehicle will continue to bounce (oscillate up and down) for some time after hitting the bump and the passengers would feel sea sick. Think SHM!

Pendulum would keep going with full amplitude for ever

Friction and air resistance reduce the amplitude until eventually the pendulum stops

What does a car shock absorber look like?
Investigate damping
Plot amplitude versus number of swings or against time. Use the graph to check for exponential decay of the amplitude using the constant ratio property.
Use excel to plot your graphs
Exponential decay of amplitude
Exponential decay of the amplitude A implies that:

Rate of decay of maximum amplitude A proportional to present value of A.

Large amplitude implies high maximum velocity, which implies greater drag.

Recall that PEMAX is proportional to A^2, so the total energy of an oscillator proportional to A^2

When the amplitude has decayed to 1/2 its original value, the energy has been reduced to 1/4 of the original input and so on.
Damping
In the “real-world” a system in SHM such as a pendulum or a mass on a spring will eventually come to rest due to “losing energy”, actually doing work against friction and air resistance!
Goldilocks and the three dampers...
What will the ride in the car be like when
1. The damping is not strong enough
2. The damping is too strong
3. The damping is just right

1. Light damping (under damping) The oscillation slowly dies away (exponential decay).
2. Heavy damping (over damping) The damping is so strong that the displaced object takes a long time to return to the equilibrium position and does not oscillate
3. Critical damping : just enough to stop the oscillation as quickly as possible often about T/4 with little or no overshoot
3 damping situations
Problems 1
1. For each situation say whether damping is, light, critical or heavy
a. A child on a swing.
b. Oil in a U shaped tube, displaced and released from equilibrium.
2. What would happen if we replaced the normal oil in a shock absorber with a more viscous oil?
3. A family car has an unloaded mass of 1000kg. The mass is supported equally by 4 springs. When fully loaded the total mass increases to 1250kg causing the springs to compress by a further 2cm. When the car hits a bump in the road it bounces. Find the period of these oscillations.
[0.64s]

Resonance
Objectives:
To Understand resonance (C)
To be able to qualitatively explain how a system behaves at resonance and on either side of resonance in terms of amplitude & phase (C)
To apply this to real world examples (A)
The devasting effects of resonance...
Simple definition :

Resonance is the tendency of a system to oscillate with a larger amplitude at some frequencies than at others.

This particular frequency is called the resonant frequency & at this point the system is said to be in resonance
Resonance
Can you link this to a child on a swing?
The pushes are an example of an applied periodic force and the swing now experiences forced oscillations.

If timed correctly the pushes take the swing higher & higher
Child on a swing and resonance
What do the different colour lines represent?
What are the effects?
* Between the displacement and applied periodic force
Examples of resonance
Resonance circus - have a go at each experiment. There is a sheet for each. Be prepared to brief the others on how each one works afterwards.
Resonance and damping
Increased damping causes the resonant frequency to fall below the natural frequency
Complete activities TAP 307-7, 8, 9
Apply knowledge
Apply this knowledge to the exam questions....
Oscillations revision activities
Options:
Tarsia (I'll print off a blank grid if you need)
Battleships (work out how to make it a revision activity)
Teach a lesson?