Simple harmonic motion Flashcards
1
Q
What is simple harmonic motion defined in terms of?
A
Acceleration and displacement
2
Q
What does an object moving with shm do?
A
- An object moving with SHM oscillates to and fro, either side of a mixed point
- The distance of the object from the midpoint is called its displacement
- There is always a restoring force pulling or pushing the object back towards the midpoint
3
Q
What is the size of the restoring force directly proportional to?
A
To the displacement i.e. if the displacement doubles, the restoring force doubles to
4
Q
What is acceleration directly proportional to?
A
As the restoring force causes acceleration towards the midpoint, we can also say the acceleration is directly proportional to displacement
5
Q
What are the conditions for SHM?
A
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
6
Q
What does the restoring force do?
A
Makes the object exchange Ep and Ek
7
Q
What does the type of potential energy depend on?
A
- 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
8
Q
What happens as the object moves towards the midpoint?
A
- 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
9
Q
What happens when the object is at the midpoint?
A
Ep = 0 Ek = maximum
10
Q
What happens at the maximum displacement (the amplitude) on both sides of the midpoint?
A
Ep = max Ek = 0
11
Q
What is the sum of the potential and mechanical energy called?
A
The mechanical energy and stays constant (as long as the motion isn’t damped)
12
Q
What is the energy transfer for on complete cycle of oscillation?
A
Ep, Ek, Ep, Ek and then the process repeats
13
Q
How does displacement, x vary?
A
Displacement, x, varies, as a cosine with a maximum value, A (the amplitude)
14
Q
How does velocity, v vary?
A
- 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
15
Q
How does acceleration, a vary?
A
- The gradient of the velocity-time graph
- It has a maximum value of omega^2A, and is in anti phase with the displacement
16
Q
What do the frequency and period not depend on?
A
The amplitude
17
Q
What is a cycle of oscillation?
A
- 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
18
Q
What is the frequency f of the SHM?
A
The number fo cycles per second (measured in Hz)
19
Q
What is the period, T?
A
The time taken for a complete cycle (in seconds)
20
Q
What is the angular frequency, omega?
A
2pif
21
Q
What is a rule of SHM?
A
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
22
Q
What are the SHM equations?
A
- For an object moving with SHM, the acceleration, a is directly proportional to the displacement, x
- 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 - 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 - 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
23
Q
What is an example of a simple harmonic oscillator?
A
- 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
24
Q
What is the formula for the period of a mass oscillating on a spring?
A
T = 2pi (m/K)^1/2
T: period of oscillation in second
m: mass in kg
k: spring constant in NM^-1
25
How do you check the formula for the period of a mass oscillating on a spring experimentally?
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
26
How can you use this set up to investigate factors which affect period?
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)
27
What is an example fo a simple harmonic oscillator?
A simple pendulum
28
How do you show that a simple pendulum is an example of SHO?
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
29
What is the formula for the period of a pendulum?
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
30
What are free vibrations?
- 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
31
When do forced vibrations happens?
- 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
32
What is driving frequency?
- 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
33
When does resonance happen?
When driving frequency = resonant frequency
34
What happens when the driving frequency approaches the resonant frequency?
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
35
What are examples of resonance?
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
36
When does damping happen?
- 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
37
What does dampening do?
- 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
38
What does critical dampening do?
Critical dampening reduces the amplitude (i.e. stops the system oscillating) in the shortest possible time
39
What do car suspension systems do?
- 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
40
How does damping affect resonance?
- 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
41
Why are structures damped?
- 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
42
How can damping be used to improve performance?
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
