Oscillations

Question 1

Define displacement of wave

Answer 1

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

Question 2

Define amplitude

Answer 2

The maximum displacement from its undisturbed position

Question 3

Define time period

Answer 3

The time taken for one complete oscillation

Question 4

Define frequency

Answer 4

The number of vibrations per second of any oscillating system

f = 1/T

Question 5

Define angular velocity, and state the equation to calculate it

Answer 5

Angular velocity is the rate of change of angle for a circulating object

ω = dθ/dt

ω = 2π/T
or ω = 2πf

Question 6

Define angular frequency, and state the equation to calculate it

Answer 6

Angular frequency is the number of cycles per unit time

It uses the same equations as angular velocity

ω = 2π/T
or ω = 2πf

Question 7

What is phase difference and what is it measured in?

Answer 7

Phase difference is how out of step two identical waves are. It is measured in degrees or radians.

Question 8

How do you calculate phase difference?

Answer 8

t/T x 360 = degrees
or
t/T x 2π = rad

Where t is the time interval between two corresponding points on the waves, and T is the time period

Question 9

State the conditions for a particle to perform simple harmonic motion

Answer 9

• the acceleration of a particle is directly proportional to its displacement
• the acceleration of the particle is directed in the opposite direction to the displacement

Question 10

What are the shapes of the displacement/time, velocity/time and acceleration/time graphs of a particle in SHM, starting from rest position

Answer 10

• displacement/time graph is a sine wave
• velocity/time graph is a cosine wave
• acceleration/time graph is a sine wave reflected in the x-axis

Question 11

What are the relationships between displacement, velocity and acceleration for a particle in SHM?

Answer 11

• the velocity is a maximum when the acceleration is zero and the displacement is zero
• the acceleration is a maximum when the displacement is at a maximum and the velocity is zero

Question 12

State the formal relationship between acceleration and displacement for a particle in SHM

Answer 12

acceleration ∝ -displacement
a ∝ -x

Question 13

What is the constant of proportionality added to the relationship, a ∝ -x, and what equation does it form?

Answer 13

Constant of proportionality = ω² or (2πf)²

a = -ω²x
or a = -(2πf)²x

This is the general equation for SHM

Question 14

Write the equation and it’s variations that can be used to calculate displacement, x, when x = 0 at t = 0

Answer 14

x = Asinθ
x = Asin(ωt)
x = Asin(2πft)

Question 15

Write the equation and it’s variations that can be used to calculate displacement, x, when x = A at t = 0

Answer 15

x = Acosθ
x = Acos(ωt)
x = Acos(2πft)

or x = Asin(2πft + φ)

Question 16

What are the equations for velocity and acceleration given by differentiating x = Asin(2πft)

Answer 16

v = 2πfAcos(2πft)
or v = ωAcos(ωt)

a = -(2πft)²Asin(2πft)
or a = -ω²t²Asin(ωt)

Question 17

State the equation for velocity containing amplitude and displacement, and hence state the equation for maximum velocity

Answer 17

v = +/- ω√(A² - x²)

Velocity is a maximum when displacement is zero

Therefore v max = +/- ωA
or v max = +/- 2πfA

Question 18

State what will happen to the period of a simple harmonic oscillator if its amplitude is changed

Answer 18

The period of a simple harmonic oscillator is independent of its amplitude. This means that the period stays the same when the amplitude is changed.

Question 19

What are the energy changes in SHM?

Answer 19

In an oscillation there is a constant interchange of energy between potential and kinetic, and if the system is undamped its total energy is constant

Question 20

What are the relationships between displacement, kinetic energy and potential energy for a system in SHM?

Answer 20

• the kinetic energy is a maximum when the displacement is zero and potential energy is zero
• the potential energy is a maximum when the displacement is at a maximum and the kinetic energy is zero

Question 21

State the equation for potential energy in a spring system

Answer 21

Ep = 1/2kx²

Question 22

State the equation for total energy

Answer 22

E total = 1/2kA²

Question 23

State the equation for kinetic energy, using the equations for E total and Ep

Answer 23

Ek = E total - Ep
Ek = 1/2kA² - 1/2kx²
Ek = 1/2k(A² - x²)

Question 24

Define the total potential energy of a system for a spring oscillating vertically

Answer 24

For a spring oscillating vertically, the total potential energy will be the sum of the elastic and gravitational potential energy of the system

Question 25

State and describe the three types of damped motion

Answer 25

Under-damped motion - there is an exponential envelope
Critical damping - no oscillation occurs, but the motion ceases in as short a time as possible
Over-damped motion - no oscillation occurs but the time to return to the equilibrium position is longer than for the critical case

Question 26

Define critical damping

Answer 26

Critical damping occurs when the motion ceases after the shortest possible time

Question 27

What is a feature of an exponential envelope?

Answer 27

In equal times the amplitude is seen to fall by equal fractions

Eg. if A1 A2 A3 etc are amplitudes after equal intervals of time then

A1/A2 = A2/A3 etc

Question 28

What is the difference between natural frequency and forcing frequency?

Answer 28

All systems have a natural frequency of oscillation, which could be SHM, a more complex harmonic motion or damped harmonic motion.
An outside agent can apply a forcing frequency of any value, f, to the system. This forcing frequency is a source of energy, passed on to the system.

Question 29

Define resonant frequency

Answer 29

When the forcing frequency is equal to the natural frequency of the system, then resonance is said to occur, and amplitude is a maximum

Question 30

State the relationships between resonance and damping

Answer 30

• as we increase damping, the amplitude of the resonant vibration decreases
• the resonant peak also becomes broader, and resonance occurs at a lower frequency