Section 7 - Further Mechanics Flashcards
1
Q
What is 360° in radians?
A
2π radians
2
Q
What is 45° in radians?
A
π/4 radians
3
Q
What is angular speed?
A
- The angle an object rotates through per unit time.
- Unit: rad/s
(angle/time)
4
Q
What is the symbol for angular speed?
A
ω
5
Q
What is the symbol for angle?
A
θ
6
Q
What is the symbol for time?
A
t
7
Q
What is linear speed?
A
- The speed at which an object is covering distance
* Units: m/s
8
Q
What is the unit for angular speed?
A
rad/s
9
Q
What is the unit for linear speed?
A
m/s
10
Q
What is the symbol for radius?
A
r
11
Q
What equation gives you the velocity of an object moving in a circular path of a radius
A
12
Q
How do you get from that equation to ω = v/r
A
13
Q
What are the two basic equations for angular speed?
A
- ω = θ / t (NOT GIVEN)
* ω = v / r
14
Q
In a cyclotron, a beam of particles spirals outwards from a central point. The angular speed of the particles remains constant.
why?
A
All the parts of the particle beam rotate through the same angle in the same time so they have the same angular speed.
15
Q
In a cyclotron, a beam of particles spirals outwards from a central point. The angular speed of the particles remains constant. The beam of particles in the cyclotrons rotates through 360° in 35μs. Explain why the linear speed of the particles increases as they spiral outwards, even though their angular speed is constant.
A
Linear speed depends on r, the radius of the circle being turned as well as ω (v = ωr). So, as r increases, so does v, even though ω remains constant.
16
Q
In a cyclotron, a beam of particles spirals outwards from a central point. The angular speed of the particles remains constant. The beam of particles in the cyclotrons rotates through 360° in 35μs. Calculate the linear speed of a particle at a point 1.5m from the centre of rotation.
A
- First calculate the linear speed:
- ω = θ / t = 2π / (35 x 10^-6) = 1.7951 x 10^5 rad/s
- Then substitute ω into v = ωr:
- v = ωr = 1.7951 x 10^5 x 1.5 = 2.6927 x 10^5 m/s
- v = 2.7 x 10^5 m/s (to 2 s.f.)
17
Q
What is angular frequency?
A
The same as angular speed.
18
Q
In circular motion, what is frequency?
A
- The number of compete revolutions per second.
* Units: rev/s or Hz
19
Q
What are the units for frequency?
A
Hertz (Hz)
20
Q
In circular motion, what is period?
A
- The time taken for a complete revolution.
* Units: s
21
Q
What is the unit for period?
A
Seconds (s)
22
Q
What is the symbol for frequency?
A
f
23
Q
What is the symbol for time period?
A
T
24
Q
What is the equation linking time period and frequency?
A
f = 1 / T
25
What are the three basic equations for angular speed?
* ω = θ / t (NOT GIVEN)
* ω = v / r
* ω = 2πf = 2π / T
26
Derive the equation that links angular speed and frequency.
• ω = θ / t
• ω = 2π / T = 2πf
or
27
What is the difference between t and T?
* t is the time
| * T is the time period
28
Are objects travelling in a circle always accelerating?
Yes, because:
• They are changing direction, so the velocity is changing constantly
• Acceleration is defined as rate of change of velocity, so the car is accelerating even if the speed isn’t changing
29
Which way is an object accelerating when it moves in a circle?
Towards the centre.
30
What is the force that causes acceleration in a circle and which way does it act?
* Centripetal force
| * It acts towards the centre of the circle
31
What is the acceleration towards the centre of a circle in circular motion called?
Centripetal acceleration
32
What is the symbol for centripetal acceleration?
a
33
What is centripetal acceleration?
* The acceleration towards the centre of a circle of an object in circular motion
* Units: m/s²
34
What are the units for centripetal acceleration?
m/s²
35
What are the two formulas for centripetal acceleration?
* a = v² / r
| * a = ω²r
36
How are these derived?
a = v² / r
a = ω²r
37
What is centripetal force (include acceleration)?
* The force that causes the centripetal acceleration and keeps an object moving in circular motion
* Units: N
38
What is the symbol for centripetal force?
F
39
What is the unit for centripetal force?
Newtons (N)
40
Why does centripetal have force (newtons first)?
Newtons 1st = constant velocity unless a force acts on it
| Travelling in a circle = changing velocity = acceleration = centripetal acceleration = must be a force causing this
41
What are the formulas for centripetal force?
* F = mv² / r
| * F = mω²r
42
How are these derived with Newtons second Law?
F = mv² / r
F = mω²r
```
F=ma
a = v² / r
a = ω²r
F = mv² / r
F = mω²r
```
43
What would happen if the centripetal force were removed?
The object would fly off at a tangent.
44
Which way does the centripetal force act?
Towards the centre of the circle.
45
Car loop the loop:
| What is the support force acting inwards at each stage?
```
At the side, (it is the same on the left side),
the support (centripetal) force is going inwards and the weight force is going downward, at 90 degrees from the support (centripetal) force, so the weight force doesn't effect the support force:
S = mv^2/r
```
At the bottom, the support force has to hold the car up against weight.
S = mv^2/r +mg
At the top, weight is pulling down, contributing to the centripetal force.
S = mv^2/r - mg
46
What is the tension at each the top and bottom?
47
What does SHM stand for?
Simple harmonic motion
48
What is the point around which simple harmonic motion occurs called?
Midpoint (equilibrium)
49
What is displacement in simple harmonic motion?
The distance of an object from the midpoint.
50
What keeps an object in simple harmonic motion?
A restoring force pulling the object towards the midpoint at all times.
51
Describe how the restoring force in SHM changes.
It is directly proportional to the displacement.
52
In SHM, if displacement doubles, what happens to the restoring force?
It also doubles.
53
Describe the acceleration in SHM.
| What is it proportional to and what is it directed towards?
It is directly proportional to the displacement and always directed towards the midpoint.
54
What is the condition for SHM (acceleration)?
An oscillation in which the acceleration of an object is directly proportional to its displacement from the midpoint, and is directed towards the midpoint.
55
Give the condition for SHM as an equation.
a ∝ -x
| Where a = acceleration and x = displacement.
56
What does the minus sign show?
| a ∝ -x
acceleration is always opposing the displacement
57
What is the symbol for potential energy?
Ep
58
What is the symbol for kinetic energy?
Ek
59
What energy exchanges occur in SHM?
Potential energy is transferred to kinetic energy and back.
60
What type of potential energy (Ep) is involved in SHM?
It depends on what is providing the restoring force.
61
For a pendulum in SHM, what type of potential energy is involved?
Gravitational
62
For masses on springs moving horizontally in SHM, what type of potential energy is involved?
Elastic
63
In SHM, what causes the transfer from Ep to Ek?
The work done by the restoring force.
64
Describe energy transfers in SHM.
NOTE: The sum of the potential and kinetic energy stays constant.
65
Describe the energy of an object at maximum displacement in SHM.
* Ep is at its maximum
| * Ek is zero
66
Describe the energy of an object at the midpoint in SHM.
* Ek is at its maximum
| * Ep is zero
67
What is the sum of potential and kinetic energy in SHM called?
Mechanical energy
68
What is mechanical energy and what happens to it in SHM?
* It is the sum of the potential and kinetic energy.
| * It stays constant (as long as the motion isn’t damped).
69
Describe the graph for Ek and Ep against time in SHM.
* Both curves are the same and are half a cycle out of phase
* Each curve is like a sine wave that has been shifted above the x-axis.
* Note that this means that the wave plateaus slightly each time it reaches the x-axis (i.e. it is not a sharp point at the bottom, but a curve)
70
Remember to practice drawing out the graph for energy in SHM.
Pg 100 of revision guide
71
Describe the graph of displacement against time in SHM.
* It is a cosine wave that goes above and below the x-axis.
* Starts at the positive maximum.
* Maximum values are A and -A.
72
Describe the graph of velocity against time in SHM.
• It is a cosine wave that goes above and below the x-axis.
• Starts at the origin, then goes down.
• Maximum values are ωA and -ωA (where ω = angular frequency and A = amplitude.)
Remember ω = 2πf
(displacement but 90 degrees (π /2) to the left)
73
Describe the graph of acceleration against time in SHM.
• It is a cosine wave that goes above and below the x-axis.
• Starts at the negative maximum.
• Maximum values are ω²A and -ω²A (where ω = angular frequency and A = amplitude.)
(Reflected displacement)
74
Why does the acceleration-time graph have minima's when velocity has maxima's?
Because it is a stationary point for velocity, meaning it's derivative (acceleration) is 0
75
Describe how the graphs for displacement, velocity and acceleration against time in SHM are related (in terms of fractions of wave cycles)
* They are of the same shape, except velocity is 1/4 of a cycle to the left of displacement and acceleration is 1/2 a cycle to the left of displacement.
* Each graph is the gradient of the previous one.
76
What can be said about the graphs for displacement and acceleration against time in SHM?
The acceleration graph is in antiphase with the displacement. (π/2 radians out of phase)
77
What are the maximum and minimum values for displacement in a displacement-time graph in SHM?
* Maximum: A
* Minimum: -A
(Where A = amplitude)
78
What are the maximum and minimum values for velocity in a velocity-time graph in SHM?
* Maximum: ωA
* Minimum: -ωA
(Where ω = angular frequency and A = amplitude)
79
What are the maximum and minimum values for acceleration in an acceleration-time graph in SHM?
* Maximum: ω²A
* Minimum: -ω²A
(Where ω = angular frequency and A = amplitude)
80
Remember to practise drawing out the graphs for displacement, velocity and acceleration against time in SHM.
Pg 100 of revision guide
81
What is a cycle in SHM?
From maximum positive displacement to maximum negative displacement and back.
82
What is frequency in SHM?
The number of cycles per second.
83
What is period in SHM?
The time taken for a complete cycle.
84
What is angular frequency?
* The same as angular speed, except in SHM.
| * Equal to 2πf.
85
In SHM, are frequency and period dependent on amplitude?
No - frequency (and period) are independent of the amplitude
86
When a pendulum swings, does the amplitude of the swing have any effect on period or frequency?
No - even if it's swings become very small the pendulum will keep ticking in regular time intervals
87
What is the defining equation of SHM?
a = -ω²x
88
How can you convert this into the defining equation of SHM:
| a ∝ -x
• Introduce a constant of proportionality
• This depends on ω
• So the equation is:
a = -ω²x
89
When will it's speed appear the fastest and slowest?
90
How do you get the the displacement equation for SHM?
91
What must be true for the displacement equation to work?
To use this equation you need to start when the pendulum is at it's maximum displacement - i.e. when t=0, x =±A.
Also you must be in radians
92
How do you get the acceleration equation for SHM?
93
How do you get maximum acceleration for SMH
94
In SHM what would a graph of acceleration against displacement look like?
95
Add flashcard on how to define angular frequency in SHM.
Do it.
96
Give the equation for acceleration in SHM.
a = -ω²x
97
Give the equation for the maximum acceleration in SHM.
a(max) = ω²A
98
What is the equation for velocity in SHM?
v = ±ωsqrt(A² - x²)
99
Why is there a plus or minus in this equation:
| v = ±ωsqrt(A² - x²)
The object moves in both directions.
| If it moves in the positive direction it is +. If it moves in the negative direction it is -.
100
What is the equation for maximum velocity in SHM?
Max velocity = ωA
101
What does a graph of Vmax against A look like?
The equation is similar to a circle equation
102
What is the equation for displacement in SHM?
x = Acos(ωt)
103
In the equation for displacement in SHM, x = Acos(ωt), at what point is t = 0?
* The object must be at maximum displacement when t = 0.
| * i.e. When t = 0, x = A.
104
What type of motion does a mass on a spring demonstrate?
Simple harmonic motion
105
When mass is pushed or pulled either side of the equilibrium position on a spring, what is exerted on it?
Restoring force
106
In a mass-spring system, what is the equation for the force that is used to displace the mass from its equilibrium position?
F = -kx
(Where k = spring constant and x = displacement)
(NOT GIVEN IN EXAM)
107
How do you get F= -kx?
108
Give the equation for the period of a mass-spring system.
T = 2πsqrt(m/k)
| Where m = mass and k = spring constant
109
How do you derive T = 2πsqrt(m/k)
110
What is a mass-spring system?
A mass on a spring that can be displaced from its equilibrium to show SHM.
111
How do atoms in a solid lattice move?
They vibrate in SHM.
112
Describe the experiment to check the formula for the period of a mass-spring system.
1) Using string, tie a trolley to a spring
2) Put masses in the trolley
3) Place a position sensor in front of the trolley and spring
4) Pull the trolley to one side by a certain amount and let go.
5) The trolley will oscillate back + forwards as the spring pulls and pushes it in each direction
6) you can measure the period, T, by getting a computer to connect to the position sensor and create a displacement-time graph from a data logger.
7) Read off the period, T, from the graph.
113
In the experiment investigating a mass-spring system, what types of potential energy are involved in the vertical spring?
Elastic and gravitational. For a horizontal spring, it would be just elastic.
114
How can you investigate the relationship between mass and Time period of a mass - spring system?
Change the mass (m) by loading the trolley with masses.
115
Describe the relationship between T and m in a mass-spring system and how this can be shown graphically.
* T ∝ √m
| * So a graph of T² against m can be plotted, which shows a straight line of positive gradient.
116
How could you change the experiment to find the relationship between spring constant and Time period?
Change the spring constant (k) by using different combinations of springs
117
Describe the relationship between T and k in a mass-spring system and how this can be shown graphically.
* T ∝ √(1/k)
| * So a graph of T² against 1/k can be plotted, which shows a straight line of positive gradient.
118
How can you investigate the relationship between Time period and amplitude of a mass -spring system?
Change the amplitude (A) by pulling the trolley across by different amounts.
119
Describe the relationship between T and A in a mass-spring system and how this can be shown graphically.
* There is no relationship.
| * So a graph of T against A can be plotted, which shows a straight horizontal line.
120
Describe what graphs can be plotted in an investigation into factors affecting a mass-spring system. What does each illustrate?
* T² against m -> Illustrates T ∝ √m
* T² against 1/k -> Illustrates T ∝ √(1/k)
* T against A -> Shows there is no relationship between T and A.
121
What factors affect the period of a mass-spring system?
* Mass
| * Spring constant
122
What type of motion does a simple pendulum show?
Simple harmonic motion
123
What examples of SHM do you need to know about?
* Mass-spring system
| * Simple pendulum
124
Describe how factors affecting SHM in a simple pendulum can be investigated.
1) Attach a pendulum to an angle sensor connected to a computer.
2) Displace the pendulum by a small angle (less than 10°) and let it go.
3) The angle sensor measures how the bob’s displacement from the rest position varies with time.
4) Use the computer to plot a displacement-time graph and read off the period, T, from it. Take the average of several oscillations to reduce percentage uncertainty.
5) Change the mass of the pendulum bob (m), amplitude of displacement (A) and length on the rod (l) independently to see how they affect the period. Plot each graph.
(NOTE: Alternatively, you could also measure the period using a stopwatch. Measure several oscillations and divide by the number to get an average.)
125
What is a pendulum made of?
* Light, stiff rod
| * Pendulum bob
126
When measuring the period of a simple pendulum, what reference point should be used to measure the start of each oscillation?
The midpoint of the swing, because:
• This is where the pendulum moves fastest, so it is more clear-cut when it reaches the midpoint
• The clamp from which it is hung may be used as a marker
127
Describe the relationship between T and l in a pendulum and how this can be shown graphically.
* T ∝ √l
| * So a graph of T² against l can be plotted, which shows a straight line of positive gradient.
128
Describe the relationship between T and m in a pendulum and how this can be shown graphically.
* There is no relationship.
| * So a graph of T against m can be plotted, which shows a straight horizontal line.
129
Describe the relationship between T and A in a pendulum and how this can be shown graphically.
* There is no relationship.
| * So a graph of T against A can be plotted, which shows a straight horizontal line.
130
What is the formula for the period of a pendulum?
T = 2π√(l/g)
| Where l = length of pendulum (m) and g = gravitational field strength (N/kg)
131
Derive T = 2π√(l/g)
132
What are the constraints of the formula for the period of a pendulum?
It only works for small angles of oscillation (up to about 10°).
133
What is a free vibration?
One where there is (theoretically) no transfer of energy to or from the surroundings.
134
Remember to revise SHM graphs.
Pgs 102-103 of revision guide.
135
How does a mass on a spring oscillate when stretched and released?
* At its resonant (natural) frequency.
* It is a free vibration, so if no energy is transferred to or from the surroundings, it will keep oscillating with the same amplitude.
136
What frequency does the sound from striking metal vibrate at?
The sound you hear is caused by vibrations at the object's natural frequency.
137
If you hit a metal object and it vibrates at it's natural frequency, what happens if it were to have free vibrations?
No energy is transferred to or from the surroundings, it will keep oscillating with the same amplitude forever. In practice this never happens.
138
Although oscillations can't happen forever, what do we call a spring vibrating in air anyway?
free vibration
139
What is a forced vibration?
When a system is forced to vibrate by a periodic external force.
140
What is the frequency of a driving force causing forced vibrations called?
Driving frequency
141
What is driving frequency?
The frequency of a driving force causing forced vibrations.
142
What happens when the driving frequency is much less than the resonant frequency?
The two are in phase since the oscillator just follows the motion of the driver.
143
What happens when the driving frequency is much greater than the resonant frequency?
The oscillator can’t keep up with the driver, so the two are completely out of phase (antiphase)
144
What is the resonant frequency of a mass-spring system?
The frequency at which a system oscillates when allowed to vibrate freely.
(CHECK THIS!)
145
What happens when driving force approaches natural frequency?
System gains more and more energy from the driving force so vibrates with a rapidly increasing amplitude.
The system is resonating.
146
What is resonance?
* When the driving frequency equals the resonant frequency
* Phase difference between the driver and oscillator is 90°
* Maximum amplitude of vibration is achieved
147
Describe how amplitude of oscillation changes with driving frequency? Explain with force, energy, phases difference
* When the driving frequency is less than the resonant frequency, the amplitude is low
* As the driving frequency approaches the resonant frequency, the amplitude increases at an increasing rate -> More energy is being transferred from the driving force
* Peak amplitude is at the resonant frequency
* As driving frequency increases above the resonant frequency, the amplitude decreases at a decreasing rate -> The driver and oscillator are completely out of phase, so less energy is transferred from the driving force.
148
Practice drawing out the set-up to investigate resonance in a mass-spring system.
Pg 104 of revision guide
149
What is the phase difference between the driver and oscillator at resonance?
90°
150
What is damping?
When an oscillating system loses energy to surroundings.
151
Give some examples of resonance.
* Organ pipe - The air resonates when driven by a motion of air in the base. Creating a stationary wave.
* Swing - resonates when it's driven by someone pushing it at it's natural frequency.
* Glass smashing - glass resonates when driven by sound wave at the right frequency.
* Radio - tuned so the electric circuit resonates at the same frequency as the radio station you want to listen to.
152
What causes damping?
Damping forces, such as air resistance.
153
Give an example of a damping force.
Air resistance
154
Is damping always unwanted?
No, sometimes systems are deliberately damped to stop them oscillating or to minimise the effect of resonance.
155
How do shock absorbers in a car suspension work?
They provide a damping force by squashing oil through a hole when compressed.
156
Is all damping the same?
No, there can be different degrees.
157
How does damping affect the amplitude of oscillation and how does the degree of damping change this?
The heavier the damping, the quicker the amplitude is reduced.
158
What is heavy and light damping
Light = long to stop oscillating. The amplitudes reduce by small amounts each period.
Heavy = Less time, amplitudes much smaller per period.
159
What is critical damping?
• Where the amplitude is reduced to equilibrium in the shortest possible time • Only one sharp displacement is seen
160
What is overdamping?
* When the damping is too heavy so the system takes longer to return to equilibrium than a critically damped system
* Only one long displacement is seen.
161
Give an example of overdamping
162
How can a pendulum on a string and a book on a string represent heavy and light damping?
163
Give two examples of when critical damping is used.
* Car suspension systems are critically damped so they don't oscillate but return to equilibrium as soon as possible.
* Moving coil meters
164
Describe the graphs of displacement against time for light damping and heavy damping.
* Light damping is like a sine wave of gradually decreasing amplitude
* Heavy damping is the same, except the amplitude decreases more sharply
165
Describe the graph of displacement against time for critical damping.
* Starts at a positive displacement
* The curve is like a gradual and very stretched sine wave until the x-axis, except it begins to flatten out before it reaches the axis
* The line then continues flat along the axis
166
How do plastic deformation effect amplitude?
reduces amplitude of oscillations the same way as damping.
As the material changes shape, it absorbs energy, so the oscillation will be smaller.
167
Does damping affect resonance?
Yes
168
How does damping affect resonance (resonant peak graph)?
* The heavier the damping, the flatter the graph of amplitude against driving frequency will be.
* This means the amplitude at the resonant frequency will be lower.
* The difference in amplitude is most marked near the resonant (natural) frequency.
169
How does damping affect resonance (in general, not in terms of the graph)?
Mark Scheme:
| Reduces sharpness of resonance.
Reduced amplitude at resonant frequency
170
Describe the set-up used to investigate how damping affects resonance.
* Mass is attached between two springs that are clamped vertically
* Vibration generator is attached to the bottom of the lower spring
* Signal generator is connected to the vibration generator
* Discs of different sizes can be attached to the mass to affect air resistance
* The amplitude of oscillation is looked at by seeing how far the mass moves
171
Give some used of damping.
* Car suspensions
* Moving coil meters
* Skyscrapers
* Soundproofing
172
Describe how damping can be used to improve sound performance.
* Sound waves reflect off walls and, at certain frequencies, cause stationary waves to be created between walls of a room.
* This causes resonance and can affect sound quality.
* Soundproofing walls in music studios absorbs the sound waves and converts their energy to heat energy. This prevents the stationary waves from forming.
173
What is an example of skyscrapers using damping?
| What type of damping is this?
