3.2 Vibrations

Question 1

What is simple harmonic motion?

Answer 1

When an object moves such that its acceleration is always directed towards a fixed point and is proportional to its distance from the fixed point (a = -ω²𝑥)

Question 2

What is the amplitude of an oscillating body?

Answer 2

The maximum displacement from the equilibrium position

Question 3

What is the phase?

Answer 3

A value of how far a point is through a wave cycle. E.g Asin(wt+E) where E is a phase constant.

Question 4

What is damping?

Answer 4

The lessening of amplitude of free oscillations due to resistive forces and reduction in energy

Question 5

What is critical damping?

Answer 5

When the resistive forces on the system are just large enough to prevent oscillations occurring at all when the system is displaced and released

Question 6

What is forced oscillations?

Answer 6

They occur when a system capable of natural oscillations is subjected to sinusoidally (sine curve-like) varying driving force

Question 7

What is resonance?

Answer 7

When the frequency of the driving force (or driving vibration) is equal to the natural frequency of the system, the amplitude of the resulting oscillations is large. This is resonance.

Question 8

How can you prove a = -ω²𝑥 ?

Answer 8

a = -Aω²cos(ωt+E)
x = Acos(ωt+E)
so a = -ω²𝑥

Question 9

What is the PE equal to in SHM for a spring

Answer 9

PE = 1/2k𝑥²
ω² = k/m
so
PE = 1/2mω²𝑥²
or 1/2mω²A²cos²(ωt+E)

Question 10

What is KE equal to in SHM?

Answer 10

KE = 1/2mv²
= 1/2mω²A² when Vmax = Aω
so
KE = 1/2mω²A²sin²(ωt+E)

Question 11

What is the sum of KE and PE?

Answer 11

KE = 1/2mω²A²sin²(ωt+E)
PE = 1/2mω²A²cos²(ωt+E)
KE + PE = 1/2mω²A²(sin²(ωt+E) +cos²(ωt+E)) = 1/2mω²A²(1)
KE + PE = 1/2mω²A²

Question 12

Why is the KE and PE always equal?

Answer 12

Energy is transferred from PE to KE and vice versa meaning no energy is lost and the energy is always equal providing there is no resistive forces.

Question 13

What are free oscillations (Natural oscillations)

Answer 13

Free oscillations occur when an oscillatory system (such as a
mass on a spring, or a pendulum) is displaced and released. They are damped
The system will resonate at its natural frequency

Question 14

What is light damping?

Answer 14

When energy is slowly removed from an oscillating system so the amplitude slowly decreases over many cycles

Question 15

What is very heavy damping?

Answer 15

When an oscillating system returns to equilibrium position slowly without oscillation (i.e. door mechanism)

Question 16

What is heavy damping?

Answer 16

When energy is quickly removed from an oscillating system so it is at rest within a few cycles.

Question 17

What is critical damping?

Answer 17

When an oscillating system returns to equilibrium very quickly but without further oscillations.

Question 18

What is the period of natural oscillation equal to for a spring?

Answer 18

T = 2π√m/k

m = mass (kg)
k = Springs constant (Nm^-1)

Question 19

What is the period of natural oscillation equal to for a Pendulum?

Answer 19

T = 2π√l/g
l = length of string
g = gravitational acceleration

Question 20

At what frequency is resonance achieved?

Answer 20

The natural frequency of a free oscillation.

Question 21

Why is ω = √k/m for a spring and why is T = 2π√m/k

Answer 21

F = k𝑥
F = mω²𝑥
so
k𝑥 = mω²𝑥

ω² = k/m
ω = √k/m

T = 2π/ω
T = 2π/√k/m = 2π√m/k

Question 22

Why is ω = √g/l for a spring and why is T = 2π√l/g

Answer 22

Force towards equilibrium when the angle is small is:
F = mgsinθ
For this system, r = L so
mω²L = mgsinθ
ω² = g/Lsinθ
Small angle approximation: sinθ = θ
ω² = g/Lθ