Chapter 17. Oscillations

Question 1

Define period

Answer 1

The time taken for one complete oscillation or vibration

Question 2

Define frequency

Answer 2

The number of complete oscillations per unit time

Question 3

What is the relationship between frequency and period?

Answer 3

Frequency, f = 1 / T , period

Question 4

What are is the unit of frequency and its equivalence?

Answer 4

Hertz, 1 Hz = 1 s^-1

Question 5

Define amplitude

Answer 5

The maximum displacement from equilibrium position

Question 6

What is meant by isochronous?

Answer 6

The ability of an oscillator to maintain a constant period despite change in amplitude

Question 7

Define simple harmonic motion

Answer 7

The motion of a particle about a fixed point such that its acceleration a is proportional to its displacement x from the fixed point and is in the opposite direction, a = -ω²x

Question 8

What is the solution for the equation of simple harmonic motion?

Answer 8

x = x₀sinωt

Question 9

What are the formulae for velocity for SHM?

Answer 9

v = v₀cosωt when x = x₀sinωt

v₀= x₀ω

V max = +/- ω * root of x max squared minus x squared

Question 10

What is the formula for acceleration?

Answer 10

a = −a0 sinωt when x = x0 sin ωt

Question 11

Define free oscillation

Answer 11

A particle is said to be undergoing free oscillations when the only external force acting on it is the restoring force.

Question 12

What are the effects of damping?

Answer 12

» the amplitude of oscillation at all frequencies is reduced
» the frequency at maximum amplitude shifts gradually towards lower frequencies
» the peak becomes flatter.

Question 13

Explain what is meant by resonance

Answer 13

Resonance occurs when the natural frequency of vibration of an object is equal to
the driving frequency, giving a maximum amplitude of vibration.

Question 14

Define period

Answer 14

The time taken to complete one oscillation

Question 15

Define frequency

Answer 15

The number of oscillations completed per unit time

Question 16

What is the relationship between frequency and period?

Answer 16

T = 1/f

Question 17

Define displacement

Answer 17

The distance from equilibrium position

Question 18

Define amplitude

Answer 18

The maximum displacement from equilibrium position

Question 19

What is meant by isochronous?

Answer 19

The oscillations maintain a constant time period

Question 20

Define simple harmonic motion

Answer 20

motion of a particle about a fixed point where the acceleration, a is proportional to the displacement x from the fixed point, and is in the opposite direction, a = - ω²x, where ω is the angular frequency, 2πf

Question 21

Why is the constant for the acceleration displacement equation for s.h.m squared?

Answer 21

To preserve the negative sign which shows that acceleration acts in the opposite direction to displacement

Question 22

Desribe the solution of the equation for simple harmonic motion

Answer 22

The solution is of the form x = x₀sin(ωt) when the particle is displaced from the equilibrium position and released from rest, where x is the displacement, x₀ is the amplitude, and conversely x = x₀cos(ωt) when the particle is at maximum displacement at time t =0.

Question 23

Describe the instantaneous velocity equation of s.h.m

Answer 23

v = -v₀sin(ωt) when x = x₀cos(ωt) and v = v₀cos(ωt) when x = x₀sin(ωt)

Question 24

What is the formula for maximum velocity from amplitude?

Answer 24

v₀ = x₀ω

Question 25

Describe the instantaneous velocity equation of s.h.m

Answer 25

a = -a₀sin(ωt) when x = x₀sin(ωt) and a = -a₀cos(ωt) when x = x₀cos(ωt)

Question 26

Derive the formula for instantaneous velocity from instantaneous displacement

Answer 26

1. x = x₀sin(ωt) and v = x₀ωcos(ωt) 2. sin²θ + cos²θ = 1 so x²/x₀² + v²/x₀²ω² = 1 3. so v² = x₀²ω² -x²ω² 4. v = +/- ω√(x₀² - x²)

Question 27

What is the formula for kinetic energy of a particle in s.h.m

Answer 27

E_k = 1/2mω²(x₀² - x²)

Question 28

Describe the kinetic energy displacement graph of a particle in s.h.m

Answer 28

A parabola with the maximum kinetic energy at x = 0 and o energy at minimum and maximum displacement.

Question 29

Derive the formual for potential energy of a particle in s.h.m

Answer 29

1. F = ma so restoring force = -mω²x 2. work done is force * displacement so E_p = (1/2)mω²x², the hald comes from the average force over the displacement, the graph is a sinusodial graph and we all know the average of sin squared is 1/2

Question 30

Describe the graph for the potential energy of a particle in s.h.m

Answer 30

A parabola with the 0 potential energy at x = 0 and maximum potential energy at minimum and maximum displacement.

Question 31

Derive the formula of the total energy and explain why it is constant

Answer 31

E_{tot = E^k + E^p, so Etot = (1/2)mω²(x0² - x²) + (1/2)mω²x², Etot = (1/2)mω²x0², it is constant as the formula clearly has constants only, the two graphs are mirror images of each other and the total energy is the sum of the two.}

Question 32

Describe free oscillations

Answer 32

oscillations which only have the restoring force as the external force, energy is not dissipated and amplitude remains constant

Question 33

Define damping and the types

Answer 33

The dissipation of energy in an s.h.m, light , heavy or over and critical

Question 34

Describe resonance

Answer 34

When the natural frequency of vibration of an object is equal to the driving frequency of the external force giving maximum displacement.

Question 35

Describe an amplitude driving frequency graph with and without damping

Answer 35

Amplitude increases with frequency until at maximum at the resonant frequency and then decreases. With damping, all amplitudes are decreased, and the top is flatter, its more for heavier damps. The frequency at maximum amplitude shifts gradually towards the lower frequencies

Question 36

How would you decrease the effect of damping on a system?

Answer 36

By increasing the mass of the system