Periodic Motion: Simple Harmonic Motion

Question 1

What is the amplitude of oscillations?

Answer 1

The max displacement of oscillating object from eqm.

Question 2

For two objects oscillating at the same frequency, phase difference is radians is?

Answer 2

phase difference = 2πΔt/T,
where Δt is the difference between successive instances when the two objects are at max displacement in same direction.

(bc 2π = 360)

In degrees = 360Δt/T

Question 3

Acceleration is greatest when gradient of velocity-time graph is greatest. This is when velocity is..
and displacement is..

Answer 3

..zero and occurs at max displacement in the OPPOSITE direction.

Question 4

What are the conditions of SHM?

Answer 4

If the acceleration of a body is always directly proportional to its displacement FROM a fixed point, and always directed TOWARDS that point, the motion is SH.

a ∝ - x a = - ω^2 x

Question 5

Time period is independent of?

Answer 5

Amplitude of oscillations.

Question 6

What’s the restoring force?

Answer 6

For any oscillating object, the resultant force acting on the object acts towards the eqm position. This is the restoring force.

Question 7

So if restoring force, F ∝ - x?

Answer 7

If F ∝ - x then a ∝ - x ∴ SHM

Question 8

Mass-spring system: when does frequency decrease? Explain fully. (2)

Answer 8

• mass increases bc increases inertia ∴ at given x slower oscillations therefore takes more time.
• weaker spring used bc restoring force at given x decreases ∴ a and v decrease ∴ takes longer time.

SHM:
a =F/m= -kx/m = -k/m x = -ω^2x

Question 9

Simple pendulum: what’s restoring force ( think weight) and ∴ what’s a?

Assumption?

Answer 9

Restoring force = -mgsinθ
∴ a = -mgsinθ/m = -gsinθ

Small angle approx sinθ = x/L

∴ a = -g/L x = -ω^2x

(at eqm T-mg = mv^2/L)

Question 10

See sheet for derivation of T eqs!!!!!!!!!!!!!!!!!!!!!!!!!!!

Question 11

Spring-mass system:
Et = Ek +Ep.
What’s Ek? What’s v?

Answer 11

Ek = Et - Ep = 1/2 k (A^2 - x^2)
= 1/2 mv^2
Bc ω^2 = k/m, above can be written as v^2 = ω^2 (A^2 - x^2)

Question 12

What are dissipative forces?
What are damped oscillations?

Answer 12

Pendulum oscillations gradually die away because of air resistance gradually reduces the total energy of the system. The forces causing the amplitude to decrease are dissipative forces.
The motion is damped if dissipative forces are present.

Question 13

Light damping?

Answer 13

Amplitude gradually decreases, reducing by same fraction each cycle.
T is independent of A, so each cycle takes same time as the oscillations die away.

Question 14

Critical damping?

Answer 14

System returns to eqm position in shortest possible time without oscillating.
Damping is just enough to stop the system oscillating after its been displaced from eqm position and released.

Question 15

Heavy damping?

Answer 15

Occurs when damping is so strong that the displaced object returns to eqm much more slowly than if the system were critically damped.
Eg a spring in thick, viscous oil.

Question 16

How can we see if the damping /amplitude has exponential decay?

Answer 16

See if it has a constant half life on the graph.

Question 17

What’s a free oscillating object?

Answer 17

A free oscillating object oscillates with a constant amplitude because there’s no friction/external force acting on it. The only forces acting on it combine to provide the restoring force.

Question 18

What’s natural frequency?

Answer 18

When a system oscillates without a periodic force being applied/oscillates freely, the system’s frequency is called it’s natural frequency.

Question 19

What’s a periodic force?

Answer 19

An external force repeatedly applied at regular intervals. System undergoes forced vibrations.

Question 20

As we increase the applied frequency from zero: (2)

1 - amplitude
2 - phase difference

Answer 20

1 - amplitude of oscillations increase until it reaches maximum amplitude at a particular frequency, and then A decreases again.
2 - phase difference between displacement and periodic force increases from zero to 1/2π to π as the f increases further.

Question 21

At max A, phase difference is 1/2π ∴ periodic force is exactly in phase with the velocity of the oscillating system, and the system is…..

Answer 21

is in resonance. The f at max A is called the resonant frequency.
The lighter the damping:
-the larger the max A becomes at resonance, and
-the closer the resonant f is to natural f of the system.

Question 22

So for an oscillating system with little to no damping at resonance:

Answer 22

the applied frequency of the periodic driving force= the natural frequency of the system

Question 23

At resonance, the periodic force acts on the system..

Answer 23

at the same point in each cycle, causing the A to increase to a max value limited only by damping.

(At max A energy supplied by the periodic force is lost at the same rate bc of the effect of damping.)

Question 24

If we plot a graph of amplitude of forced oscillations against driving frequency, we see a very large peak at..

Answer 24

the natural frequency/resonant frequency.

Question 25

Increased damping causes above graph to

Answer 25

Have a lower peak bc lower max A due to energy losses from system, less sharp peak/broader peak due to damping, reduced frequency (bc longer to complete each oscillation so resonant f gets further and further from natural f)

Question 26

Resonance eg 1: Barton’s pendulums:
Pendulums along horizontal string, pendulum B is on one side and DEFH on other side, all of different length strings.
Pendulum E has same length as B. B is oscillated. What is observed and why?

Answer 26

Effect of oscillating motion transferred to rest of pendulums along horizontal string. E oscillates in resonance with B bc it’s subjected to forced oscillations of the same f as its natural f.

Question 27

Resonance eg 2: Bridge oscillations:
If a bridge span is not fitted with —–, it can be made to oscillate at resonance if the bridge span is subjected to a suitable periodic force.

Answer 27

dampers

Can also: stiffen structure,
increase mass of bridge,
restrict number of pedestrians,
redesign to change natural f