6. Further Mechanics and Thermal Physics Flashcards
1
Q
What are Newton’s three laws of motion?
A
- An object stays at rest or in uniform motion unless acted upon by a force
- The rate of change of momentum of an object is proportional to the resultant force acting upon it
- When two objects interact, they exert equal and opposite forces on each other
2
Q
Symbol for momentum?
A
p
3
Q
Unit for momentum?
A
kg ms-1
4
Q
How is F=ma derived from Newton’s second law?
A
- F ∝ Δp / Δt
- F ∝ mΔv/Δt
- F=ma or F=kma
5
Q
When is F=ma true?
A
When mass is constant
6
Q
How would you generally give NII?
A
F = Δ(mv)/Δt
7
Q
How would you give NII if mass remains constant?
A
F = mΔv/Δt or F=ma
8
Q
How would you give NII if mass is changing?
A
F = vΔm/Δt
where Δm/Δt is mass change per second
9
Q
What is the symbol for angular displacement?
A
θ
10
Q
What is the unit for angular displacement?
A
rad
11
Q
What is the symbol of angular velocity?
A
ω
12
Q
What is the unit for angular velocity?
A
rad s-1
13
Q
If a wheel takes T seconds to rotate once, what angle will it turn through each second?
A
2π/T radians
14
Q
What will the frequency of a rotation of a wheel that takes T seconds to rotate once be?
A
f=1/T
15
Q
What is angular displacement given by?
A
θ = 2πt / T or θ = 2πft
16
Q
How do we get the equation ω = 2πf?
A
- circumference = 2πr
- time for one rotation = distance travelled / velocity = 2πr / v or 2π / ω
- so 2πr / v = 2π / ω therefore v = r ω
- also θ = 2πft and ω = θ / t
- ∴ ω = 2πf
17
Q
What is centripetal acceleration given by?
A
a = v2 / r
= ω2 r
18
Q
How do we know there must be a centripetal force when a bike wheel is rotating?
A
It is accelerating as it is changing direction, so there must be a force
19
Q
Why does kinetic energy stay constant when a bike wheel rotates?
A
- W=Fd in direction of force
* because F is at 90 degrees to v, no work is done
20
Q
What is the equation for centripetal force?
A
F = mv2 / r
= mrω2
21
Q
Is centripetal force a type of force?
A
No, it is a description of a force - it is a force that causes the acceleration to be towards the centre
22
Q
When is the centripetal force required increased?
A
- mass increased
- speed increased
- or radius decreased
23
Q
For questions about cars going over bumps/hills, what will the fastest speed that the car can travel over the hill be given by?
A
mg = mvmax2 / r or vmax = √gr
24
Q
For questions about cars going round bends, what will the maximum friction be given by?
A
Frictionmax = m vmax2 / r
25
For questions about cars on banked tracks, what can tanθ be given by?
tanθ = v2 / gr
26
What are oscillations?
Wobbling about a centre point
27
What is equilibrium?
The centre point
28
What is amplitude (A)?
The maximum displacement form centre point
29
What happens in TSM if there is no friction?
The amplitude is constant (free oscillation)
30
How does free oscillation occur?
There is no friction, so the amplitude is constant
31
What is the period (T)?
Time for one complete oscillation
32
What is the frequency (f)?
The number of oscillations per second (Hz)
33
How is simple harmonic motion defined?
As oscillating motion where the acceleration is proportional to the displacement in the opposite direction to the displacement
34
What is acceleration proportional to in SHM?
Directly proportional to displacement from the equilibrium point (a ∝ x)
35
What are the conditions for SHM?
* acceleration ∝ displacement from the equilibrium point (a∝x)
* displacement and acceleration are in opposite directions (a∝ -x) OR acceleration is always towards equilibrium point
36
What is displacement (x)?
The distance from the equilibrium position
37
What is Simple Harmonic Motion a type of?
Oscillation
38
What is phase difference, ɸ?
The fraction of an oscillation between the position of two oscillating objects
39
What is phase difference given by?
Δt/T x2π
40
What is angular frequency, ω?
The rate of change of angular position (given by 2πf)
41
What is the key equation for SHM?
a = -ω²x
42
What does it mean, that an oscillator in SHM is an isochronous oscillation?
The period of the oscillation is independent of the amplitude
43
What does isochronous mean?
Occupying equal time
44
Graphically, how is the velocity of a pendulum given?
The gradient of a displacement-time graph
45
Graphically, how is the acceleration of a pendulum given?
By the gradient of a velocity-time graph
46
In SHM, does T depend on A?
No
47
When using the equation a=-(2πf)²x, what must be remembered?
2πft must be in radians
48
How to prove equation for T=2π√m/k?
F = -kx and F = ma
ma = -kx (rearrange for a)
SHM: a = -(2πf)²x
(2πf)² = k/m
1/2π . T = √m/k
T = 2π √m/k
49
How to prove equation for T=2π√l/g?
(R) Tcosθ = mg
(R) -Tsinθ = ma
Tsinθ / Tcosθ = ma/mg
-tanθ = a/g
small angles tanθ = sinθ (displacement/length = x/l)
-x/l = a/g
(2πf)² = g/l
T/2π = √l/g
T = 2π√l/g
50
Where is a fiducial marker usually placed in the pendulum experiment?
Equilibrium position
51
Which two equations can be used to determine the displacement of a simple harmonic oscillator?
* x = A sinωt
| * x = A cosωt
52
What is the difference between waves and oscillations in terms of energy?
Waves transmit energy, oscillations do not
53
For the equations used to determine the displacement on a simple harmonic oscillator, when is each used?
* x = A sinωt for when oscillator begins at equilibrium position
* x = A cosωt for when oscillator begins at amplitude position
54
In SHM, where does maximum velocity occur?
At the equilibrium position, with the oscillator being stationary at the amplitude points
55
In SHM, where does the maximum acceleration occur?
At the amplitude points, and is 0 when the oscillator is at equilibrium position
56
What is the velocity of the oscillator given by?
v = ±ω √A²-x²
57
How is the equation for maximum velocity derived from v = ± ω√A²-x²?
Maximum velocity occurs at the equilibrium position, where x=0, so vmax = ωA
58
What is the equation for maximum velocity?
ωA
59
In SHM, what forms of energy are involved?
Kinetic and potential
60
When does maximum Ek occur in SHM?
At the equilibrium point - where velocity is a maximum
61
When does maximum Ep occur in SHM?
At the amplitude positions, where displacement is at a maximum
62
In SHM, is total energy conserved?
Yes
63
What is damping?
The process by which the amplitude of the oscillations decreases over time
64
Why might the amplitude of oscillations decrease over time?
Energy loss to resistive force e.g. drag, friction
65
What happens in SHM if there's no friction?
Free oscillation
66
What happens if frictional forces are acting in SHM?
The amplitude decreases
67
How does the energy-displacement graph for energy in SHM look?
* Ek - 'sad' parabola(max. at displacement = 0)
* Ep - 'happy' parabolamax at either side where displacement is max.)
* total energy - straight line in line with max. values for Ep and Ek
68
What are the three types of damping?
* critical
* heavy
* light
69
What is critical damping?
Object returns to equilibrium in shortest possible time and does not overshoot
70
What is heavy damping?
Where damping is so strong that object takes a long time to return to equilibrium - oscillation does not happen
71
Does oscillation occur with heavy damping?
No
72
How does damping occur in a car suspension system?
* includes spring and damper
| * damper provides almost critical damping so that oscillation dies away quickly after going over a bump
73
How does a displacement-time graph look?
Oscillations start large and then decrease by the same fraction each cycle
74
What is light damping?
Where T stays the same, and amplitude decreases by the same fraction each cycle (exponentially)
75
What happens to T in light damping?
Stays the same
76
What happens to amplitude in light damping?
Decreases by the same fraction each cycle (exponentially)
77
When does light damping occur?
Naturally (e.g. pendulum oscillating in air)
78
What happens to amplitude when heavy damping occurs?
Amplitude decreases dramatically
79
Example of when heavy damping may occur?
Pendulum oscillating in water
80
Example of when critical damping may occur?
Pendulum oscillating in treacle
81
What happens to amplitude in critical damping?
The object stops before one oscillation is completed
82
What is natural frequency?
Frequency at which a SHM oscillator vibrates when no force is applied
83
What is periodic force?
Regular pushes at the right times to make an SHM oscillator to swing high
84
When does a system undergo forced oscillation?
When a periodic force is applied
85
What is forced oscillation?
The oscillation of a system when a periodic force is applied
86
What happens if applied frequency = natural frequency?
* amplitude of oscillations becomes very big
* phase difference between displacement and periodic force changes to π/2
* periodic force is now in phase with velocity
87
What is it called when applied frequency = natural frequency?
The system is resonating
88
What happens, in terms of applied and natural frequency, when damping is light?
Applied/driving frequency of periodic force = natural frequency of system
89
What happens, in terms of applied and natural frequency, when damping is heavy?
Resonant frequency is slightly lower than natural frequency
90
What factors affect resonance?
* when oscillating mass ↑, natural frequency decreases
* when springs are weaker, natural frequency decreases
* damping limits oscillations
91
What is the link between free oscillation and natural frequency?
When an object is in free oscillation, it vibrates at its natural frequency
92
What is driving frequency the same as?
Applied frequency
93
When does resonance occur?
When the driving frequency of the external force is the same as the natural frequency of the object
94
What will happen when a system is resonating and there is no damping?
The amplitude will continue to increase until the system fails
95
What will happen when a system is resonating and damping increases?
The amplitude will decrease at all frequencies, and the maximum amplitude occurs at a lower frequency
96
What is the problem with finding the natural frequency of a system when there is no damping?
It is very hard to do
97
How can the resonance of an object be investigated experimentally?
* suspend mass between two springs attached to oscillation generator
* ruler placed parallel with mass-spring system to record amplitude
* driver frequency of generator slowly increased from zero - reaching max. amplitude when driver frequency reaches natural frequency of system
* amplitude of oscillation decreases as frequency is increased further
98
When can a comparatively weak vibration in one object cause a strong vibration in another?
Through resonance
99
How is g derived from the pendulum experiment?
* T = 2π √L/g - square both sides
* Graph T²-L
* gradient = 4π²/g
* so g = 4π²/gradient
100
How to verify Hooke's law from the mass-spring experiment?
* T² = 4π² x m/k
* graph T²-m
* gradient = T²/m = 4π²/k
* draw graph F=kx and see if gradient = 4π²/m
