Simple harmonic motion

Question 1

What is an oscillation?

Answer 1

The repetitive variation of time t of the displacement x of an object about an equilibrium position

Question 2

What is the equilibrium position?

Answer 2

The position where there is no resultant force acting on an object

Question 3

What is displacement?

Answer 3

The distance of a point on a wave from its equilibrium position

Question 4

What is amplitude?

Answer 4

The maximum value of the displacement on either side of the equilibrium position

Question 5

What is wavelength?

Answer 5

The length of one complete oscillation measured from the same point on two consecutive waves

Question 6

What is period?

Answer 6

The time interval for one complete oscillation

Question 7

What is frequency?

Answer 7

The number of oscillations per second

Question 8

What is angular frequency?

Answer 8

The rate of change of angular displacement with respect to time

Question 9

State the equations for angular frequency:

Answer 9

1. ω = 2π / T
2. ω = 2πf

Question 10

State the equations for period:

Answer 10

1. T = 1 / f
2. T = 2π / ω

Question 11

What is phase difference?

Answer 11

A measure of how much a point or a wave is in front or behind another

Question 12

How do you calculate phase difference?

Answer 12

For waves of the phase frequency:
1. P-d = (Δt / T) * 2π
2. P-d = Δλ / λ) * 2π

Question 13

What is simple harmonic motion?

Answer 13

A specific type of oscillation that occurs when the acceleration of an object is proportional to its displacement from a fixed point, but in the opposite direction

Question 14

What are the conditions for simple harmonic motion?

Answer 14

1. The oscillations are periodic
2. There is a fixed equilibrium point
3. The displacement, velocity and acceleration of the object change continually
4. The restoring force directed towards the fixed point
5. The magnitude of the restoring force is proportional to the displacement
6. The restoring force causes the acceleration

Question 15

What are some examples of SHM?

Answer 15

1. The pendulum of a clock
2. A child on a swing
3. A mass on a spring
4. Guitar string vibrating
5. The vibrations of a bowel

Question 16

How do you calculate the acceleration of an object in SHM?

Answer 16

a = -(⍵^2)x
Conclusions:
1. Acceleration is a maximum when the displacement is at a maximum or at the amplitude
2. The minus sign shows that acceleration and displacement are always in opposite directions

Question 17

How do you calculate the displacement of an object in SHM?

Answer 17

For an object that begins oscillating from the equilibrium position:
x = x-0sin(⍵t)

Question 18

When does an object in SHM have maximum displacement?

Answer 18

The displacement is a maximum at its amplitude or when sin(⍵t) = 1 or -1

Question 19

How do you calculate the speed of an object in SHM?

Answer 19

For an object that begins oscillating from the equilibrium position:
v = v-0cos(⍵t)

Question 20

When does an object in SHM have a maximum speed?

Answer 20

The maximum speed occurs at the amplitude of motion of the object

Question 21

What is the equation relating the speed and displacement of an object in SHM?

Answer 21

v = ±⍵√(x-0^2-x^2)
Conclusion:
1. When an oscillator has a greater amplitude, it has to travel a greater distance in the same amount of time and hence has a greater speed

Question 22

When is maximum speed achieved in SHM?

Answer 22

Maximum speed is reached at the equilibrium position; the SHM speed equation becomes v-0 = ⍵x-0

Question 23

What does the displacement-time graph look like for SHM?

Answer 23

The displacement-time graph is a sine curve

Question 24

What does the velocity-time graph look like for SHM?

Answer 24

1.The velocity-time graph is a cosine curve
2. It is the gradient of the displacement-time graph
3. It is 90° out of phase with the displacement-time graph

Question 25

What does the acceleration-time graph look like for SHM?

Answer 25

1. The acceleration-time graph is a negative sine curve 2. It is the gradient of the velocity-time graph 3. It is 90° out of phase with the velocity-time graph

Question 26

How does energy change in a horizontal mass-spring system?

Answer 26

In a horizontal mass-spring system, there is an interplay between elastic potential energy and kinetic energy: 1. EP is at a maximum when the spring is stretched or compressed to its maximum displacement from the equilibrium position 2. EK begins to increase once the mass is released and and it accelerates towards the equilibrium position 3. EK then decreases while EP increases pass the equilibrium position

Question 27

How does energy change in a simple pendulum system?

Answer 27

In a simple pendulum system, there is an interplay between gravitational potential energy and kinetic energy: 1. GPE is at a maximum at the point of maximum displacement from the equilibrium position 2. EK increases as the pendulum is released and moves towards the equilibrium position 3. At the equilibrium position, its velocity at a maximum and hence EK at a maximum 4. Once the pendulum swings past the equilibrium position, EK decreases and GPE increases

Question 28

How can you calculate the total energy of a system in SHM?

Answer 28

1. E = 1/2m⍵^2x-0^2 2. E = EP + EK where EP = elastic potential energy (1/2kx^2) + gravitational potential energy (mgh)

Question 29

What are the features of an energy-displacement graph for half an SHM oscillation?

Answer 29

1. EP is always at a maximum at the amplitude, but is always zero at the equilibrium position; this is seen as a 'U' shaped curve 2. EK is always at a minimum at the amplitude, but is always a maximum at the equilibrium position; this is shown by a 'n' shaped curve 3. The total energy is represented by a horizontal straight line above the curves

Question 30

What is damping?

Answer 30

Damping is the reduction in energy and amplitude of oscillations due to resistive forces on the oscillating system

Question 31

What happens to frequency in damped oscillations in SHM?

Answer 31

The frequency of damped oscillations does not change as the amplitude decreases

Question 32

What is the restoring force?

Answer 32

The force that brings the oscillator back to the equilibrium position

Question 33

What are the three types of damping?

Answer 33

1. Light damping - the amplitude decreases exponentially over time until the system comes to a stop 2. Heavy damping - it takes a long time to return to its equilibrium position without oscillating 3. Critical damping - the system will return to its equilibrium position in the shortest time possible without oscillating

Question 34

What does light damping look like on a displacement-time graph?

Answer 34

There are many oscillations with a gradually decreasing amplitude over time; this is shown by the decreasing eight of the curves in both the positive and negative displacements; the amplitude decreases exponentially

Question 35

What does critical damping look like on a displacement-time graph?

Answer 35

The system does not oscillate, meaning the displacement falls to 0 straight away; the graph has a fast decreasing gradient when the oscillator is first displaced until it reaches the x-axis; when the oscillator reaches the equilibrium position, the graph is a horizontal line at x = 0 for the remaining time

Question 36

What does heavy damping look like on a displacement-time graph?

Answer 36

There are no oscillations and so the graph does not pass 0; the graph has a slow decreasing gradient from the the oscillator is first displaced until it reaches the x-axis; the oscillator reaches the equilibrium position after a long period of time, after which the graph remains a horizontal line for the remaining time

Question 37

What is resonance?

Answer 37

The phenomenon when a system is forced to oscillate at its natural frequency by the driving frequency, giving the system more energy, causing the system to oscillate with a maximum amplitude

Question 38

What is the natural frequency?

Answer 38

The frequency at which the system is allowed to oscillate freely

Question 39

What is the driving frequency?

Answer 39

The frequency is an external force that is applied to a system to make it oscillate