Simple Harmonic Motion

Question 1

Define the amplitude of an oscillating object

Answer 1

The amplitude is the maximum displacement of the oscillating object from the equilibrium

Question 2

Give the conditions required for an oscillating object to be described as freely oscillating

Answer 2

• Constant amplitude

- no frictional forces are present

Question 3

Define the time period T, of an oscillating object

Answer 3

The time period is the time for one complete cycle of oscilations

Question 4

Define the frequency of oscillations, f

Answer 4

The frequency of oscillations is the number of cycles per second made by an oscillating object

Question 5

Give the unit for frequency

Answer 5

Hz

Question 6

Give the equation linking time period T, to frequency f

Answer 6

T = 1 / f

Question 7

Give the angular frequency (velocity) , for and object with time period T and give its units

Answer 7

ω = 2π / T = 2πf

Units : rads⁻¹

Question 8

Give the phase difference for two objects oscillating at the same frequency

Answer 8

Phase difference (in radians) = 2πΔt / T

where Δt is the time between successive instants when the two objects are at maximum displacement in the same direction

Question 9

For an object oscillating at a constant frequency, describe the differences of the graphs of:

i) displacement vs. time
ii) velocity vs. time
iii) acceleration vs. time

Answer 9

i) If the object is started with maximum positive displacement, after T/4 it will be at minimum displacement, and at T/2 it will be at maximum negative displacement
ii) the object will start with 0 velocity, but will be at maximum negative velocity at T/4 and at maximum positive velocity at 3T/4
iii) the object will start with maximum negative acceleration, and will have 0 acceleration at times T/4 and 3T/4

Question 10

Define simple harmonic motion

Answer 10

oscillating motion in which the acceleration is proportional to, but always in the opposite direction to the displacement

Question 11

Give the acceleration for an object moving with simple harmonic motion

Answer 11

a = - (2πf)²x = - ω²x

where x is the displacement

Question 12

Give the relationship between the time period, T, and the amplitude of the oscillations

Answer 12

The time period is independent of the amplitude of the oscillations

Question 13

Give the relationship between the maximum displacement and the amplitude of an oscillating object

Answer 13

x(max) = ±A

Question 14

Describe an experiment to compare simple harmonic motion with circular motion

Answer 14

Place a simple pendulum above a turntable with a bob on it in front of a projector. Given that the initial maximum displacement is equal to the radius of the turntable, and that the turntable has the same time period as the pendulum, the shadows of the balls on the screen will stay together as they move

Question 15

For a ball on top of a turntable rotating with a constant frequency in front of projector, give the equation for the component of the acceleration of the ball which is parallel to the screen (aₓ)

Answer 15

aₓ = acosθ = - (2πf)²rcosθ = - (2πf)²x

Question 16

For an object at displacement x = +A at t = 0, if it starts with zero velocity, give the equation for its displacement at time t

Answer 16

x = Acos(2πft)

Question 17

What determines the frequency of oscillation of a loaded spring?

Answer 17

The mass of the load and the strength of the spring

Question 18

Give the effect on the time period of adding extra mass to an oscillating spring

Answer 18

It would increase the time period (because the inertia of the system would increase so at a given displacement, the load would be slower than if the mass had not been added)

Question 19

Give the effect on the time period of using weaker springs for the same mass load

Answer 19

It would increase the time period (because the restoring force on the load at any given displacement would be less so the trolley’s acceleration and speed would be less)

Question 20

Give the equation for the restoring force of a loaded spring

Answer 20

ΔT = - kx

where k is the spring constant, x is the displacement and T is the change in tension of the spring

Question 21

Give the equation for the acceleration of a loaded spring in terms of its resolving force

Answer 21

a = restoring force / mass = - kx / m

Question 22

Give the equation for the acceleration of a loaded spring showing that it moves in simple harmonic motion

Answer 22

a = - (2πf)²x

Question 23

Rearrange (2πf)² = k / m to give the time period of oscillation when the spring constant and mass of the loaded spring is known

Answer 23

(2πf)² = k / m
f = (1 / 2π )√(k / m)
T = 1 / f
T = 2π √(k / m)

Question 24

When timing oscillations, what would you use to show the centre of oscillations?

Answer 24

A fiducial marker

Question 25

For a simple pendulum, give the equation for the time period, T

Answer 25

T = 2π √(L / g) | where L is the length of the pendulum

Question 26

As the bob of a simple pendulum passes through the equilibrium, the tension directly upwards is Tₓ. Give the resultant force of the bob at this instant

Answer 26

Tₓ - mg = mv² / L

Question 27

Give the acceleration for a simple pendulum when the bob is at angle θ

Answer 27

a = F / m = -mgsinθ / m = -gsinθ

Question 28

For a freely oscillating object, give the equation for the potential energy at displacement x, and describe the shape of the graph when it oscillates from amplitude -A to +A

Answer 28

Eᴘ = ½kx² | It is parabolic

Question 29

For a freely oscillating object, give the equation for the kinetic energy at displacement x, and describe the shape of the graph when it oscillates from amplitude -A to +A

Answer 29

Eᴋ = Eᴛ - Eᴘ = ½k(A² - x²) | It is an inverted parabola

Question 30

Define dissipative forces

Answer 30

The force causing the amplitude to decrease by dissipating the energy of the system to the surroundings as thermal energy

Question 31

How is the motion of oscillations said to be if a dissipative force is present?

Answer 31

Damped

Question 32

Define light damping

Answer 32

The amplitude of oscillation gradually decreases, reducing by the same fraction each cycle

Question 33

Define critical damping

Answer 33

Where the damping is enough to stop the system oscillating after it has been displaced and released from equilibrium. The oscillating object returns to equilibrium with the shortest possible time without overshooting

Question 34

Define heavy damping

Answer 34

Where the damping is so strong that the displaced object returns to equilibrium much more slowly than if the system is critically damped. No oscillating motion occurs

Question 35

Explain how the components of a car's suspension system works

Answer 35

A suspension spring is present to reduce the forces of impact from jolts. An oil damper is fitted with each spring to prevent the chassis from oscillating long after the force of impact

Question 36

Define a periodic force

Answer 36

A force which is applied at regular intervals

Question 37

Define natural frequency

Answer 37

The frequency at which a system oscillates when not subjected to a periodic force

Question 38

Define forced oscillations

Answer 38

When a periodic force is applied to an oscillating system, the response depends on the frequency of the periodic force. The system undergoes forced oscillations

Question 39

Define resonance

Answer 39

When the applied frequency is equal to the natural frequency of the oscillating system the amplitude of oscillations becomes very large.

Question 40

At resonance, what is the phase difference between the displacement and the periodic force?

Answer 40

½π

Question 41

At resonance, what is the phase relationship between the periodic force and the velocity of the oscillating system?

Answer 41

It is exactly in phase

Question 42

What effect does damping have on the amplitude of an oscillating system in resonance?

Answer 42

The lighter the damping, the larger the amplitude becomes

Question 43

For a system in resonance, as the applied frequency becomes increasingly larger than the natural frequency of the oscillating system, what happens to: i) the amplitude of oscillations? ii) the phase difference between the displacement and the periodic force?

Answer 43

i) the amplitude of oscillations decreases more and more | ii) the phase difference increases from ½π until the displacement is π radians out of phase with the periodic force

Question 44

Explain Barton's pendulum

Answer 44

- Multiple simple pendulums of different length are set up on a piece of wire. A driver pendulum has the same length as one of the other pendulums. - The driver pendulum is displaced and allowed to oscillate perpendicular to the plane of the pendulums at rest. - The effect of the oscillation is transmitted along the support thread, subjecting the others to forced oscillations. - The pendulum with the length the same as the driver pendulum will begin to resonate since it has the same natural frequency. - The response of each other pendulum depends on how close its length is to the driver pendulum