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aqa a-level physics > Option C - Engineering Physics (Rotational Dynamics) > Flashcards

Option C - Engineering Physics (Rotational Dynamics) Flashcards

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1
Q

What is required to make something start or stop moving?

A

A force.

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2
Q

What is inertia?

A

How much an object resists a change in velocity.

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3
Q

In linear and rotational systems, what is inertia described as?

A
  • Linear - Mass of object

* Rotational - Moment of inertia

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4
Q

What is the moment of inertia?

A

A measure of how difficult it is to rotate an object or change its rotational speed.

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5
Q

What two factors does moment of inertia depend on?

A
  • Mass

* Distance from axis of rotation

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6
Q

What is the symbol for moment of inertia?

A

I

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7
Q

What are the units for moment of inertia?

A

kgm²

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8
Q

What is the equation for the moment of inertia of a point mass?

A

I = mr²

Where:
• I = Moment of inertia (kgm²)
• m = Mass (kg)
• r = Distance from the axis of rotation (m)

(NOTE: Not given in exam, but I = Σmr² is given)

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9
Q

How is the moment of inertia of an extended object (i.e. not a point mass) found?

A

Adding up the individual moments of inertia of each point mass that makes up the object.

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10
Q

What is the equation for the moment of inertia of an extended object?

A

I = Σmr²

Where:
• I = Moment of inertia (kgm²)
• m = Mass (kg)
• r = Distance from the axis of rotation (m)

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11
Q

Does an object have the same moment of inertia regardless of which point it is rotated about?

A

No, it varies.

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12
Q

How can the moment of inertia of a system be found?

A

Adding the individual moments of inertia of the different objects (e.g. adding together the moment of inertia of a wheel and a reflector on it)

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13
Q

Do you need to learn specific moments of inertia for different shapes?

A

No, they will be given during the exam.

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14
Q

a) Calculate the moment of inertia of a 750g bike wheel, which has radius of 31.1cm. The moment of inertia for a hollow cylinder is I = mr².
b) A 20.0g reflector is attached to the wheel 6.0cm in from the outer edge. Assuming the reflector behaves like a point mass, calculate the new moment of inertia of the wheel.

A

a) I = 0.75 x 0.311² = 0.0725 kgm²

b) I = 0.0725 + 0.0200 x (0.311 - 0.06) = 0.0738 kgm²

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15
Q

What is the equation for the kinetic energy of a rotating object?

A

Ek = 1/2 Iω²

Where:
• Ek = Kinetic energy (J)
• I = Moment of inertia (kgm²)
• ω² = Angular speed (rad/s)

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16
Q

A dance adds a 60.0g mass to each end of her twirling baton. The baton is uniform, 70cm long and has a mass of 150g. Assume the added masses act as point masses. Calculate the rotational kinetic energy of the baton as she spins it about its centre at an angular speed of 1.1 rad/s. The moment of inertia for a rod of length L about its centre is I = 1/12 mL².

A

Overall moment of inertia for the object:
• I = I(rod) + Σmr² = 1/2 mL² + 2 x m x (L/2)²
• I = 1/12 x 0.15 x 0.7² + 2 x 0.06 x 0.035²
• I = 0.0208 kgm²
Kinetic energy:
• Ek = 1/2 Iω² = 1/2 x 0.0208 x 1.1² = 0.0125 = 0.013J (to 2s.f.)

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17
Q

What is angular displacement?

A

The angle through which a point has been rotated.

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18
Q

What is angular velocity?

A

The angle a point rotated through per second (vector quantity).

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19
Q

What is the unit for angular displacement?

A

rad

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20
Q

What is the symbol for angular displacement?

A

θ

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21
Q

What is the unit for angular velocity?

A

rad/s

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22
Q

What is the symbol for angular velocity?

A

ω

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23
Q

What is the equation that defines angular velocity?3

A

ω = Δθ / Δt

Where:
• ω = Angular velocity (rad/s)
• θ = Angular displacement (rad)
• t = Time (s)

(NOTE: Not given in exam!)

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24
Q

What is angular speed?

A

The magnitude of the angular velocity.

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25
Q

What is angular acceleration?

A

The rate of change of angular velocity.

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26
Q

What are the units for angular acceleration?

A

rad/s²

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27
Q

What is the symbol for angular acceleration?

A

α

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28
Q

What is the equation that defines angular acceleration?

A

α = Δω / Δt

Where:
• α = Angular acceleration (rad/s²)
• ω Angular velocity (rad/s)
• t = Time (s)

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29
Q

What is meant by the equations for motion for uniform linear acceleration?

A

They are the sugar equations converted into rotational terms.

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30
Q

When do the equations for rotational motion apply?

A

When the angular acceleration is uniform.

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31
Q

What is the angular equivalent of s?

A

θ

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32
Q

What is the angular equivalent of u?

A

ω₁

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33
Q

What is the angular equivalent of v?

A

ω₂

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34
Q

What is the angular equivalent of a?

A

α

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35
Q

What are the 4 equations of motion for uniform angular acceleration?

A
  • ω₂ = ω₁ + αt
  • ω₂² = ω₁² + 2αθ
  • θ = ω₁t + 1/2 αt²
  • θ = 1/2 (ω₁ + ω₂)t
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36
Q

A figure skater initially at rest begins to spin with uniform angular acceleration. After 2.5 revolutions, she has an angular velocity of 4.9 rad/s. Calculate her angular acceleration.

A
  • α = ?
  • ω₁ = 0 rad/s
  • ω₂ = 4.9 rad/s
  • θ = 2.5 x 2π = 15.7 rad
  • ω₂² = ω₁² + 2αθ
  • α = (ω₂² - ω₁²) / 2θ = 0.76 rad/s
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37
Q

How can you convert from an angular displacement in revolutions to an angular displacement in radians?

A

Multiple by 2π.

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38
Q

How can angular velocity be found from a angular displacement against time graph?

A

It is the gradient at any given point.

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39
Q

For constant angular acceleration, what does the graph of angular displacement against time look like? What does this show?

A
  • It is the shape of a stretched y = x² graph.
  • This shows that displacement is directly proportional to t².

(See graph at bottom of page 218)

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40
Q

For constant angular acceleration, what does the graph of angular velocity against time look like?

A

Straight line of positive gradient

See graph pg 219 of revision guide

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41
Q

For increasing angular acceleration, what does the graph of angular velocity against time look like?

A

Curved line of increasing gradient

See graph pg 219 of revision guide

42
Q

For decreasing angular acceleration, what does the graph of angular velocity against time look like?

A

Curved line of decreasing gradient

See graph pg 219 of revision guide

43
Q

How can you find angular acceleration from a graph of angular velocity against time?

A

It is the gradient.

44
Q

How can you find angular displacement from a graph of angular velocity against time?

A

It is the area under the line.

45
Q

What is torque?

A

The turning effect of a force.

46
Q

What is a couple?

A

A pair of forces that cause no linear motion, but cause an object to turn.

47
Q

What is the linear equivalent of a torque?

A

Either a moment or a force.

48
Q

What is the difference between a torque and a moment?

A

They are the same, but torque is used when the object is rotating, while a moment is used when the object is in equilibrium.

49
Q

What is the symbol for torque?

A

T

50
Q

What is the unit for torque?

A

Nm

51
Q

In linear equivalents of angular equations, what replaces torque?

A

Force

52
Q

What are the two equations for torque?

A 
T = F x r
T = I x α
Where:
• T = Torque (Nm)
• F = Force (N)
• r = Perpendicular distance from from axis of rotation to point of applied force (m)
• I = Moment of inertia (kgm²)
• α = Angular acceleration (rad/s²)
53
Q

Four 100g masses are suspended from the axle of a wheel. The perpendicular distance from the point of the weight to the centre of the axis of rotation is 0.15m. When the masses are released, the wheel spins with an angular acceleration of 1.3 rad/s. Calculate the moment of inertia of the wheel. Friction is negligible.

A
  • T = Fr = mgr = 4 x 0.10 x 9.81 x 0.15 = 0.5886 Nm
  • T = Iα
  • So: I = T/α = 0.5886/1.3 = 0.452 = 0.45 kgm² (to 2 s.f.)
54
Q

What is the equation for work done relative to torque?

A

W = Tθ

Where:
• W = Work some (J)
• T = Torque (Nm)
• θ = Angular displacement (rad)

55
Q

What is the equation for power relative to torque?

A

P = Tω

Where:
• P = Power (W)
• T = Torque (Nm)
• ω = Angular velocity (rad/s)

56
Q

Louise applies a torque of 0.2Nm to turn a doorknob 90° with an angular speed of 3.1 rad/s. Calculate the work done and the power exerted by Louise to turn the doorknob.

A
  • The doorknob is turned π/2 radians.
  • W = Tθ = 0.2 x π/2 = 0.314 = 0.3J (to 1 s.f.)
  • P = Tω = 0.2 x 3.1 = 0.62 = 0.6W (to 1 s.f.)
57
Q

What is frictional torque?

A

The torque in an opposing direction to motion in a rotating object. It must be overcome.

58
Q

In real life, does a machine convert all of its power to cause an object to rotate?

A

No, some of the power must be used to overcome frictional torque.

59
Q

A cog has a moment of inertia of 0.0040 kgm² and a diameter of 20.0cm. A force of 0.070 N acts at the edge of the cog in the direction of the motion of the cog at that point, causing it to accelerate. Find the power needed to overcome the frictional torque at the point that the cog has an angular velocity of 120 rev/min, if the angular acceleration at that instant is 1.25 rad/s.

A

1) Calculate net torque on the cog:
T(net) = Iα = 0.0040 x 1.25 = 0.0050 Nm
2) Find the applied torque:
T(applied) = Fr = 0.070 x 0.0100 = 0.0070 Nm
3) Find the frictional torque:
T(net) = T(applied) - T(frictional)
T(frictional) = T(applied) - T(net)
T(frictional) = 0.0070 - 0.0050 = 0.0020Nm
4) Calculate the power needed to overcome frictional torque:
P = Tω = 0.0020 x (120 x 2π / 60) = 0.0251 = 0.025W (to 2 s.f.)

60
Q

A wheel has four 0.10kg masses suspended from it. The four masses are released. Just before they hit the ground, the masses have velocity 1.70 m/s and the wheel has 0.73 J of rotational kinetic energy, having turned through 0.90 radians. There is 0.10 Nm of frictional torque acting upon the system. Calculate the height at which the masses were initially suspended above the ground.

A
  • The GPE lost is equal to the kinetic energy gained by the masses AND the wheel, plus the work done to overcome the frictional torque.
  • mgh = 1/2 mv² + Ek(rotational) + Tθ
  • mgh = 1/2 x 0.40 x 1.70² + 0.73 + 0.10 x 0.90 = 1.398
  • h = 1.398 / (0.40 x 9.81) = 0.36 m
61
Q

What is a flywheel and what is it used for?

A
  • A heavy wheel with a high moment of inertia in order to resist changes to its rotational motion.
  • It is used to store energy.
62
Q

What does a flywheel’s high moment of inertia imply?

A

Once it is spinning, it is hard to make it stop spinning (high angular momentum).

63
Q

What is the name for making a flywheel spin faster?

A

Charging

64
Q

When charging a flywheel, what is causing be change?

A

A torque is being turned into rotational kinetic energy.

65
Q

How is a flywheel kept spinning and fully charged?

A

Just enough power is continuously input.

66
Q

How is energy taken from a flywheel?

A

When extra energy is needed in a machine, the flywheel decelerates, transferring some of its kinetic energy to another part of the machine.

67
Q

What are flywheels designed to store as much energy as possible called?

A

Flywheel batteries

68
Q

What 3 factors affect the energy stored by a flywheel?

A

1) Mass
2) Angular velocity
3) Shape

69
Q

How does mass of a flywheel affect the energy stored and why?

A
  • The higher the mass, the higher the energy stored
  • This is because the higher the mass, the higher the moment of inertia (I), so the rotational kinetic energy is increased (Ek = 1/2 Iω²)
70
Q

How does angular velocity of a flywheel affect the energy stored and why?

A
  • The higher the angular velocity, the higher the energy stored
  • This is because the higher the angular velocity, the higher the rotational kinetic energy (Ek = 1/2 Iω²)
71
Q

How does the shape of a flywheel affect the energy stored and why?

A
  • Making the flywheel spoked (if the mass is kept constant) or making the centre thinner than the edges
  • This is because the moment of inertia (I) is increased (I = Σmr²), so the rotational kinetic energy is increased (Ek = 1/2 Iω²)
72
Q

Why can you not increase the mass, shape and angular velocity of a flywheel too much?

A
  • It is impractical to have a giant, heavy flywheel taking up too much space in your machine
  • If you increase the centrifugal force too much, it can start breaking the flywheel apart
73
Q

What does a flywheel lose energy to?

A
  • Air resistance

* Friction between the flywheel and nearing

74
Q

How are a flywheel’s energy losses limited?

A
  • Lubrication -> To reduce fruit on between bearings and the wheel.
  • Levitated with superconducting magnets -> So there is no contact between the bearing and the wheel.
  • Operating in vacuums or inside sealed cylinders -> To reduce the drag from air resistance.
75
Q

What is the application of flywheels, except storing energy?

A

Smoothing torque and angular velocity

76
Q

What are engine torque and load torque?

A
  • Engine torque - The torque exerted by a machine’s engine

* Load torque - The torque due to resistance forces that a machine must oppose to be useful

77
Q

When might a flywheel be used to smooth torque and angular velocity?

A
  • When a force supplied to a system can vary (e.g. an engine that only kicks in intermittently, so the flywheel keeps the angular velocity of rotating components constant)
  • When the force that the system has to exert has to exert varies (e.g. when the load torque is too high or when the engine torque is too high)
78
Q

How can a flywheel be used to smooth torque and angular velocity when the force supplied to a system varies?

A

They use each spurt of power to charge up and then deliver the stored energy smoothly to the rest of the system.

79
Q

How can a flywheel be used to smooth torque and angular velocity when the load torque is higher than the engine torque?

A

The flywheel decelerates, releasing some of the energy to top-up the system.

80
Q

How can a flywheel be used to smooth torque and angular velocity when the engine torque is higher than the load torque?

A

The flywheel accelerates and stores the spare energy until needed.

81
Q

Give some examples of systems where a flywheel is used.

A
  • Potter’s wheel
  • Regenerative braking
  • Power grid
  • Wind turbine
  • Riveting machine
82
Q

How is a flywheel used in a potter’s wheel?

A
  • A potter’s wheel is controlled by a foot pedal, making it hard to apply a constant force to it
  • Flywheel is used to keep the speed of the wheel constant
83
Q

How is a flywheel used in regenerative braking?

A
  • When the brakes are hit, a flywheel is engaged.
  • The flywheel charges up with the energy being lost from braking.
  • When the vehicle is ready to accelerate again, the flywheel’s energy is used.
84
Q

How is a flywheel used in a power grid?

A
  • When lots of electricity is used in an area, the electricity grid sometimes cannot meet the demand.
  • Flywheels can be used to provide the extra energy needed whilst backup power stations are started up.
85
Q

How is a flywheel used in wind turbines?

A

Flywheels can be used to store excess power on windy days and during off-peak times, and to give power on days without wind.

86
Q

What is a riveting machine?

A

A machine that presses and fixes two materials together.

87
Q

How is a flywheel used in riveting machines?

A
  • An electric motor charges up the flywheel, which then rapidly transfers a burst of power as the machine presses down on the rivet and fixes two sheets of material together
  • This is useful because it stops rapid changes of power in the electric motor, which could cause it to stall, and means that a less powerful motor can be used
88
Q

What are some advantages of flywheels?

A

1) Efficient
2) Last a long time without degrading
3) Short recharge time
4) React and discharge quickly
5) Environmentally friendly (don’t rely on chemicals to store energy)

89
Q

What are some disadvantages of flywheels?

A

1) Larger and heavier than other storage methods
2) Wheel can break apart at high speed. Protective casing for this adds weight.
3) Energy lost through friction.
4) If used in moving objects, they can oppose changes in direction, which can cause problems for vehicles.

90
Q

What is the unit for angular momentum?

A

Nms

or kgm²/s

91
Q

What is the equation for angular momentum?

A

Angular momentum = Iω

Where:
• Angular momentum (Nms)
• I = Moment of inertia (kgm²)
• ω = Angular velocity (rad/s)

92
Q

What does the law of conservation of angular momentum state?

A

When no external forces are applied, the total angular momentum of a system remains constant.

93
Q

How can the law of conservation of angular momentum be written?

A

I(initial) x ω(initial) = I(final) x ω(final)

94
Q

A dis has a moment of inertia I and is rotation at an angular velocity of 4 rad/s. A second identical disc that is not spinning is placed on top of the spinning disc, where it is held in place and begins to spin. Calculate the angular velocity of the combined discs as they spin together at the same speed. Frictional losses are negligible.

A 
Before:
• Angular momentum = I₁ω₁ + I₂ω₂
After:
• Angular momentum = (I₁ + I₂)ω
Therefore:
• I₁ω₁ + I₂ω₂ = (I₁ + I₂)ω
• ω = (I₁ω₁ + I₂ω₂) / (I₁ + I₂)
Since I₁ = I₂,
• ω = (I x 4 + I x 0) / (2 x I) = 4 / 2 = 2 rad/s
95
Q

What change is seen when a spinning ice skater tucks her arms in and why?

A
  • She spins faster
  • Because the moment of inertia is decreased, so the angular velocity must increase in order to conserve angular momentum
96
Q

An ice skater is spinning with her arms out at an angular velocity of 13 rad/s. With her arms out, her moment of inertia is 3.5 kgm². She then tucks in her arms, changing her moment of inertia to 1.2 kgm². Calculate her angular velocity in revolutions per second as she spins with her arms tucked in.

A
  • I₁ω₁ = I₂ω₂
  • ω₂ = I₁ω₁ / I₂
  • ω₂ = 3.5 x 13 / 1.2 = 37.91 rad/s = 6.03 revolutions/s = 6.0 revolutions per second (to 2 s.f.)
97
Q

What is angular impulse?

A

The change in angular momentum.

98
Q

What are the two equations for the angular impulse?

A

Angular impulse = Δ(Iω)
Angular impulse = TΔt

Where:
• Angular impulse (Nms)
• I = Moment of inertia (kgm²)
• ω = Angular velocity (rad/s)
• T = Torque (Nm)
• Δt = Time (s)
99
Q

When does Δ(Iω) = TΔt apply?

A

When there is a constant torque applied.

100
Q

A spanner, initially at rest, has a constant torque of 0.3 Nm applied to it for 2 seconds. Calculate the angular impulse acting on the spanner and the angular velocity of the spanner at the end of the 2 seconds. The moment of inertia of the spanner is 0.2 kgm².

A
  • Δ(Iω) = TΔt
  • Iω(final) - Iω(initial) = TΔt
  • ω(final) = (TΔt + Iω(final)) / I = (0.3 x 2 + 0.2 x 0) / 0.2 = 3 rad/s

aqa a-level physics (14 decks)

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