Further Mechanics and Thermal Physics

Question 1

What is angular displacement?

Answer 1

If a wheel takes T seconds to rotate once, it will turn through an angle of 2π/T rad each seconds.
The frequency of each rotation f = 1/T.
Therefore, the angular displacement:
θ = 2πt/T or θ = 2πft

Question 2

What is angular velocity?

Answer 2

w = θ/t = 2π/T = 2πf
for one complete rotation:
time = distance / velocity = 2πr/v = 2π/w
therefore, w = v/r

Question 3

What is centripetal acceleration?

Answer 3

A bike wheel rotates with constant speed but is constantly changing direction.
W = Fd but the F is at right angles to the velocity therefore no work is done and no energy is transferred.
However, the velocity is constantly changing direction, therefore the bike is always accelerating.
This is called centripetal acceleration.
a = v²/r = w²r

Question 4

What is centripetal force?

Answer 4

Since the bike is accelerating, there must be force.
This is called the centripetal force.
F = mv²/r = mw²r

Question 5

The centripetal force increases if …

Answer 5

F = Δp/Δt
Δp increases i.e.
-mass increases
-speed increases
Δt decreases i.e.
-radius decreases

Question 6

Vertical circle?

Answer 6

AT TOP
F = mg + T but CF = mv²/r 
therefore, T = mv²/r - mg
AT BOTTOM
F = T - mg but CF = mv²/r
therefore, T = mv²/r + mg

Question 7

Cars going over bumps/hills/bridges?

Answer 7

The car is supported by a force S.
When v = 0, S = mg.
As v increases, the car has a greater tendency to continue in a straight line.
This is because S decreases.
F = mg - S but CF = mv²/r
therefore, S = mg - mv²/r
As v increases, a greater centripetal force is required to keep the car moving in a circle. 
This happens by S decreasing.

Question 8

Cars going round bends?

Answer 8

The centripetal force is provided by the friction of the tyres on the road.
The car has a certain v(max) before it begins to slip on the road because the friction is not enough to provide the centripetal force.
friction(max) = mv(max)²/r

Question 9

Conical circle/pendulum?

Answer 9

T(H) = Tsinθ
T(V) = Tcosθ
The forces are balanced vertically, therefore:
T(V) = mg
Tcosθ = mg
The resultant force, F = T(H) = Tsinθ
But this is the CF, therefore, Tsinθ = mv²/r
therefore, tanθ = v²/rg

Question 10

Banked tracks?

Answer 10

A car is travelling round a bend and the track is banked at an angle, θ.
If there’s no friction on the tyres, the CF is provided by the horizontal component of R.
F = Rsinθ (also equal to mv²/r)
mg is balanced by the vertical component of R so:
mg = Rcosθ
therefore, tanθ = v²/rg

Question 11

What is the equilibrium position?

Answer 11

The lowest point in an oscillating motion.

Question 12

What is displacement, x?

Answer 12

The distance from the equilibrium position.

Question 13

What is amplitude, A?

Answer 13

The maximum displacement.

Question 14

What is period, T?

Answer 14

The time taken to complete one full oscillation.

Question 15

What is frequency, f?

Answer 15

The number of oscillations per unit time.

Question 16

What is phase difference?

Answer 16

The fraction of an oscillation between the position of two oscillating objects.
Δt/T x 2π

Question 17

What is angular frequency, w?

Answer 17

The rate of change of angular position.

2πf

Question 18

What is an oscillating object?

Answer 18

An oscillating object moves repeatedly in one way, then in the opposite direction, through its equilibrium position.
The displacement of the object from its equilibrium position continually changes throughout motion.

Question 19

What do the graphs of x, v and a look like?

Answer 19

x: cos graph
v: flipped sin graph
a: flipped cos graph

Question 20

What is the gradient of the x-t graph?

Answer 20

The v-t graph.
The magnitude of v is greatest when the gradient of the x-t graph is greatest i.e. x = 0
The magnitude of v is 0 when the gradient of the x-t graph is 0 i.e. x = max

Question 21

What is the gradient of the v-t graph?

Answer 21

The a-t graph.
The magnitude of a is greatest when the gradient of the v-t graph is greatest i.e. v = 0 and x = max
The magnitude of a is 0 when the gradient of the v-t graph is 0 i.e. v = max and x = 0

Question 22

What is simple harmonic motion?

Answer 22

A type of oscillation where the acceleration of the oscillator is directly proportional to the displacement from the equilibrium position, in the opposite direction to the displacement.
a ∝ -x
An oscillator in SHM is an isochronous oscillation, so the period of the oscillation is independent of the amplitude.

Question 23

What is the equation for acceleration in SHM?

Answer 23

The constant of proportionality depends on the time period, T of the oscillation.
The shorter the time period, the faster the oscillation, therefore the larger the acceleration at any given displacement. Therefore, the larger the constant.
Therefore, a = -w²x
a(max) = w²A

Question 24

What is the equation for displacement in SHM?

Answer 24

x = Asinwt (if oscillator begins at equilibrium position)
x = Acoswt (if oscillator begins at amplitude position)

Question 25

What is the equation for velocity in SHM?

Answer 25

v = ±w√A² - x²
v(max) = wA

Question 26

What is the equation for a mass-spring system?

Answer 26

F = -kx if Hooke's law is obeyed
but F = ma 
therefore ma = -kx
a = (-k/m)x so SHM
but a = -(2πf)²x 
therefore (2πf)² = k/m so f = √k/m / 2π
but T = 1/f therefore T= 2π√m/k

Question 27

What is the equation for a simple pendulum?

Answer 27

Resolving vertically:
Tcosθ = mg
Resolving horizontally:
-Tsinθ = F
-Tsinθ = ma
therefore -tanθ = a/g
but for small angles tanθ = sinθ = x/l
therefore, -x/l = a/g hence a = -(g/l)x
but a = -(2πf)²x 
therefore (2πf)² = g/l so f = √g/l / 2π
but T = 1/f therefore T= 2π√l/g

Question 28

What energy transfer occurs in SHM?

Answer 28

If there’s no friction, free oscillation occurs.
The energy is constantly changing between Ek and Ep.
But the total energy remains constant i.e. energy is always conserved.
The max Ek occurs at equilibrium, where v = max.
The max Ep occurs at the the amplitude position, where x = max.
RECALL GRAPH

Question 29

What is damping?

Answer 29

The process by which the amplitude of the oscillations decreases over time. This is due to energy loss due to resistive forces such as drag or friction.
Successive waves become smaller.
However, the frequency always remains the same.

Question 30

What is light damping?

Answer 30

The time period, T stays constant.

The amplitude decreases by the same fraction each cycle.

Question 31

What is heavy damping?

Answer 31

The damping is so strong, that the object takes a long time to return to equilibrium.
Oscillation does not occur.

Question 32

What is critical damping?

Answer 32

The object returns to equilibrium in the shortest time possible and does not overshoot.

Question 33

What is an example of damping?

Answer 33

A car suspension system includes a spring and a damper. The damper provides almost critical damping so that oscillation dies away quickly after going over a bump.

Question 34

What is a free vibration?

Answer 34

The total energy stays the same over time therefore, the amplitude of the vibration is constant.
No external forces are applied and the object is oscillating at its natural frequency.
This is a theoretical idea because in real systems, energy is dissipated to the surroundings and the amplitude decays to 0 (damping).

Question 35

What is a forced vibration?

Answer 35

A periodic force is applied to an object, which causes it to oscillate at a particular frequency

Question 36

What is resonance?

Answer 36

If applied frequency = natural frequency:
-The amplitude of the oscillations becomes very big.
-Phase difference between the displacement and periodic force changes to π/2.
NB: Below resonance, they are in phase and about resonance, the phase difference is π.

Question 37

What is the effect of damping on resonance?

Answer 37

The resonant frequency is f0.
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 max amplitude will occur at a lower frequency.

Question 38

Examples of resonance?

Answer 38

• A swing. Initially, the motion is slow and the swing doesn’t reach its maximum potential. Once it reaches its natural frequency of oscillation, a gentle push helps it maintain that amplitude of swing all throughout due to resonance.
• Barton’s Pendulums. The driver pendulum is displaced and released, and the other pendulums undergo forced oscillation as the effect of the oscillation is transported along the wire. The pendulum which is the same length as the driver has the same time period, and thus the same natural frequency. Therefore, this pendulum oscillates with a larger amplitude.
• Bridges. In the absence of damping, a crosswind or many people walking across the bridge, in step with each other apply a periodic force equal to the natural frequency of the bridge span. This can cause resonance, and the bridge can collapse.