Simple harmonic motion

Question 1

What is simple harmonic motion defined in terms of?

Answer 1

Acceleration and displacement

Question 2

What does an object moving with shm do?

Answer 2

1. An object moving with SHM oscillates to and fro, either side of a mixed point
2. The distance of the object from the midpoint is called its displacement
3. There is always a restoring force pulling or pushing the object back towards the midpoint

Question 3

What is the size of the restoring force directly proportional to?

Answer 3

To the displacement i.e. if the displacement doubles, the restoring force doubles to

Question 4

What is acceleration directly proportional to?

Answer 4

As the restoring force causes acceleration towards the midpoint, we can also say the acceleration is directly proportional to displacement

Question 5

What are the conditions for SHM?

Answer 5

An oscillation in which the acceleration of an object is directly protectional to tis displacement form the midpoint, and is directed towards the midpoint
a alpha -x

Question 6

What does the restoring force do?

Answer 6

Makes the object exchange Ep and Ek

Question 7

What does the type of potential energy depend on?

Answer 7

1. The type of potential energy (Ep) depends now hat it is that is providing the restoring force
   - This will be gravitational Ep for pendulums
   - This will be elastic Ep (elastic stored energy_ for masses on springs moving horizontally

Question 8

What happens as the object moves towards the midpoint?

Answer 8

• The restoring force does work on the object
• And so transfers some Ep to Ek
• When the object is moving away from the midpoint, all that Ek is transferred back to Ep again

Question 9

What happens when the object is at the midpoint?

Answer 9

Ep = 0
Ek = maximum

Question 10

What happens at the maximum displacement (the amplitude) on both sides of the midpoint?

Answer 10

Ep = max 
Ek = 0

Question 11

What is the sum of the potential and mechanical energy called?

Answer 11

The mechanical energy and stays constant (as long as the motion isn’t damped)

Question 12

What is the energy transfer for on complete cycle of oscillation?

Answer 12

Ep, Ek, Ep, Ek and then the process repeats

Question 13

How does displacement, x vary?

Answer 13

Displacement, x, varies, as a cosine with a maximum value, A (the amplitude)

Question 14

How does velocity, v vary?

Answer 14

• The gradient of the displacement-time graph
• It has the maximum value of omegaA (where omega is the angular frequency of the oscillation and is a quarter of a cycle in front of the displacement

Question 15

How does acceleration, a vary?

Answer 15

• The gradient of the velocity-time graph

- It has a maximum value of omega^2A, and is in anti phase with the displacement

Question 16

What do the frequency and period not depend on?

Answer 16

The amplitude

Question 17

What is a cycle of oscillation?

Answer 17

1. From maximum positive displacement (e.g. maximum displacement to the right) to maximum negative displacement (e.g. maximum displacement to the left) and back again is called a cycle of oscillation

Question 18

What is the frequency f of the SHM?

Answer 18

The number fo cycles per second (measured in Hz)

Question 19

What is the period, T?

Answer 19

The time taken for a complete cycle (in seconds)

Question 20

What is the angular frequency, omega?

Answer 20

2pif

Question 21

What is a rule of SHM?

Answer 21

In SHM, the frequency and period are independent of the amplitude (i.e. constant for a given oscillation) so a pendulum clock will keep ticking in regular time intervals even if its swing becomes very small

Question 22

What are the SHM equations?

Answer 22

1. For an object moving with SHM, the acceleration, a is directly proportional to the displacement, x
2. The constant of proportionality depend on omega, and he acceleration is always in the OPPOSITE direction form the displacement (so there is a minus sign)
   a = -omega^2 x
   Maximum acc: amax =omega^2A
3. The velocity is positive if the object is moving in one direction snd negative if it’s moving in the opposite direction - thats why ± sign
   v = ±omega (A^2 - x^2)^1/2
   Maximum speed = omegaA
4. The displacement varies with time according to the equation
   x = Acos(omegat)
   -To use this equation you need to start timing when the pendulum is at its maximum displacement i.e when t=0 and x=A

Question 23

What is an example of a simple harmonic oscillator?

Answer 23

• A mass on a spring
  1. When the massis pushed to the left or pulled to the right of the equilibrium position.l, there is a force exerted on it
• The size of this force: F=-kx
• Where K is the spring constant (stiffness) of the spring in Nm^-1 and x is the displacement in meters

Question 24

What is the formula for the period of a mass oscillating on a spring?

Answer 24

T = 2pi (m/K)^1/2
T: period of oscillation in second
m: mass in kg
k: spring constant in NM^-1

Question 25

How do you check the formula for the period of a mass oscillating on a spring experimentally?

Answer 25

1. Set up equipment as shown (spring, clamp, positions sensors, to computer, mass) - Because the spring in this experiment is hung vertically the Ep is both elastic and gravitational - For the horizontal spring susie, shown above, the potential energy is just elastic 2. Pull the masses down a set amount, this will be your initial amplitude. Let the masses go 3. The masses will now oscillate with SHM 4. The position sensor measures the displacement of the mass over time 5. Connect the position sensor to a computer and create a displacement time graph 6. Read off T from the graph

Question 26

How can you use this set up to investigate factors which affect period?

Answer 26

1. Change the mass, m, by loading more masses, onto the spring (T^2 alpha m) 2. Change the spring stiffness constant, k, by using different combinations of springs (T^2 alpha 1/k) 3. Change the amplitude, A, by pulling the masses down by different distances (T doesn't depend on amplitude)

Question 27

What is an example fo a simple harmonic oscillator?

Answer 27

A simple pendulum

Question 28

How do you show that a simple pendulum is an example of SHO?

Answer 28

1. Attach a pendulum to an angle sensor connected to a computer 2. Displace the pendulum from its rest position by a small angle (less than 10 degrees) and let it go. The pendulum with oscillate with SHM 3. The angle sensor measures hoe the bob's displacement from the rest position varies with time 4. You can sue the the computer to plot a displacement-time groan and read off the period, T from it - Make sure you calculate the average period over several oscillations to reduce the percentage uncertainty in your measurement 5. Change the mass of the pendulum bob, m, the amplitude of the displacement, A, and the length of the rod, l, independently to see how they affect the period, T - You can also do this experiment by hanging the pendulum from a clamp and timing the oscillation using a stop watch - Use the clamp stand as a reference point so it is easy to tell when the pendulum has reached the mid-point of its oscillation

Question 29

What is the formula for the period of a pendulum?

Answer 29

T = 2pi (l/g)^1/2 (this formula only works for small angle of oscillation - up to about 10 degrees from the equilibrium point T: period of oscillation in seconds l: length of pendulum (between pivot and centre of mass of bob in m) g: gravitational field strength in Nkg^-1 -T does not depend on m -T^2 alpha l -T does not depend on A

Question 30

What are free vibrations?

Answer 30

- No transfer of energy to or from the surroundings 1. If you stretch and release a mass on a spring, it oscillates at its resonant frequency 2. If no energy is transferred to or from the surrounding, it will keep oscillating with the same amplitude forever 3. In practice this never happens, but a spring vibration in air is called a free vibration anyway

Question 31

When do forced vibrations happens?

Answer 31

- When there is an external driving force 1. A system can be force to vibrate by a periodic external force 2. The frequency of this force is called the driving frequency

Question 32

What is driving frequency?

Answer 32

- If the driving frequency is much less than the resonant frequency then the two are in phase: the oscillator just follows the motion of the driver - But, if the driving frequency is much greater than the resonant frequency, the oscillator won't be able to keep up, you end up with the driver completely out of phase with the oscillator - At resonance, the phase different between the driver and oscillator is 90 degrees

Question 33

When does resonance happen?

Answer 33

When driving frequency = resonant frequency

Question 34

What happens when the driving frequency approaches the resonant frequency?

Answer 34

The system gains more and more energy from the driving force and so vibrates with increasing amplitude and when this happens the system is resonating

Question 35

What are examples of resonance?

Answer 35

1. Organ pipe: the column of air resonates setting up a stationary wave in the pipe 2. Swing: A swing resonants if it's driven by someone pushing it at its resonant frequency 3. Glass smashing: A glass resonant when driven by a sound wave a the right frequency 4. Radio: a radio is tuned so the electric circuit resonates at the same frequency as radio broadcasts - Armies deliberately march 'out of step' when they cross a bridge. This reduces the risk of the bridge resonating and breaking apart

Question 36

When does damping happen?

Answer 36

- When energy is lost to the surroundings 1. In practice any oscillating system loses energy to its surroundings 2. This is usually down to fictional forces like air resistance 3. These are called damping forces 4. Systems are often deliberately draped to stop them oscillating or to minimise the effect of resonance - Shock absorbers in a car suspension provide a damping force by squashing oil through a hole when compressed

Question 37

What does dampening do?

Answer 37

- The degree of damping can vary from light damping (where the damping force is small) to overdamping 1. Damping reduces the amplitude of the oscillation over time 2. The heavier the damping the quicker the amplitude is reduced to zero

Question 38

What does critical dampening do?

Answer 38

Critical dampening reduces the amplitude (i.e. stops the system oscillating) in the shortest possible time

Question 39

What do car suspension systems do?

Answer 39

- Car suspension systems and moving coil meters are critically damped so that they don't oscillate but return to equilibrium as quickly as possible - Systems with even heavier damping are over damped - They take longer to return to equilibrium that a critically damped system - Plastic deformation of ductile materials reduces the amplitude of oscillations in the same way as damping - As the material changes shape, it absorbs energy, so the oscillation will be smaller

Question 40

How does damping affect resonance?

Answer 40

- Lightly damped systems have a very sharp resonance peak - Their amplitude only increases drastically when the driving frequency is very close to the resonant frequency - Heavily damped systems have a flatter response - Their amplitude doesn't;t increase very much near the resonant frequency and they are not as sensitive tot eh driving frequency

Question 41

Why are structures damped?

Answer 41

- Structures are damped to avoid being dragged by resonance | - A skyscraper may sue a very tall skyscraper use a vert large pendulum to damp oscillations caused by strong winds

Question 42

How can damping be used to improve performance?

Answer 42

1. Loudspeakers in a room create sound waves in the air 2. These reflect off the walls of the room, and at certain frequencies stationary sound waves are created between the walls of the room 3. This causes resonance and can affect the quality of the sound: some frequencies are louder than they should be - Places like recording studios use soundproofing on their walls which absorbs the sound energy and convert it into heat energy