5.3 - Oscillations

Question 1

Considering a spring oscillating, give the definitions of the following:
Displacement
Amplitude
Period
Frequency

Answer 1

● Displacement - distance from the equilibrium position (vector)
● Amplitude - maximum displacement
● Period - time taken for a complete oscillation
● Frequency - number of oscillations per second

Question 2

State the equation relating angular frequency and time period.

Answer 2

⍵ = 2π/T

Where ⍵ = angular frequency, T = time period

Question 3

What are the conditions for SHM?

Answer 3

● Acceleration must be directly proportional to displacement and in the opposite direction:
(a ∝ -𝑥)
● It must act towards equilibrium.

Question 4

What are the two main examples of systems which undergo SHM?

Answer 4

1. A mass-spring system
2. A pendulum

Question 5

What is the constant of proportionality linking acceleration and displacement?

Answer 5

• ⍵²

Question 6

What is 𝑥 as a function of t and ⍵?

Answer 6

𝑥 = Acos⍵t or 𝑥 = Asin⍵t

(where A is amplitude)

Question 7

True or false: velocity is maximum when displacement is maximum.

Answer 7

False.
The velocity is minimum at the amplitude of oscillation, as the object changes direction.
Velocity is maximum when the object passes through the equilibrium position.

Question 8

How can you calculate the maximum speed using ⍵ and A?

Answer 8

vₘₐₓ = ⍵A

Question 9

If the following graph shows displacement against time, what would the velocity-time graph look like?

Answer 9

Question 10

If the following graph shows displacement against time, what would the acceleration-time graph look like?

Answer 10

Question 11

Draw the graph for potential energy and kinetic energy against displacement for a SHM system.

Answer 11

Question 12

What is damping?

Answer 12

Damping is the process by which the amplitude of the oscillations decreases over time. This is due to energy loss to resistive forces such as drag or friction.

Question 13

With a diagram to help, explain the difference between light damping, heavy damping and critical damping.

Answer 13

Light damping occurs naturally (e.g. pendulum oscillating in air), and the amplitude decreases exponentially (but time period remains constant as A and T are independent). When heavy damping occurs (e.g. pendulum oscillating in water) the amplitude decreases dramatically. In critical damping (e.g. pendulum oscillating in treacle) the object is stopped in as short a time as possible without overshooting equilibrium.

Question 14

What is the difference between free and forced oscillations?

Answer 14

When an object oscillates without any external forces being applied, it oscillates at its natural frequency. This is known as free oscillation. Forced oscillation occurs when a periodic driving force is applied to an object, which causes it to oscillate at a particular frequency.

Question 15

What is resonance?

Answer 15

When the driving frequency of the external force applied to an object is the same as the natural frequency of the object, resonance occurs. This is when the amplitude of oscillation rapidly increases, and if there is no damping, the amplitude will continue to increase until the system fails. As damping is increased, the amplitude will decrease at all frequencies, and the maximum amplitude occurs at a lower frequency.

Question 16

Describe an experimental technique to investigate the resonance of an object

Answer 16

● Suspend a mass between two springs attached to an oscillation generator and use a ruler parallel with the spring-mass system to record the amplitude.
● Increase the frequency of the generator slowly so that the amplitude increases, reaching maximum amplitude when the driver frequency is the same as the natural frequency of the system (after which, increasing the frequency will decrease the amplitude).
● Since drag force due to the air damps the system, the amplitude should not continue to increase until the point of system failure.
● To increase accuracy, the system can be filmed and the amplitude value recorded from video stills, as it can be difficult to determine this whilst the mass is oscillating.