Engineering physics- Rotational dynamics

Question 1

What does every mass have?

Answer 1

Any mass has a resistance to a change in velocity when subjected to a force, this is called its inertia. Therefore, the larger the inertia of a mass, the larger the force required to change its velocity by a certain amount.

Question 2

What is the moment of inertia for an isolated mass?

Answer 2

A measure of how difficult it is to alter an object’s rotational speed:

I = mr²

I = moment of inertia (kg m2)
m = mass (kg)
r = distance from axis of rotation (m)

Question 3

What is the moment of inertia of an extended object?

Answer 3

I = ∑mr²

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

Question 4

If a small mass m is added to a rod at radius r from the axis of rotation, what is the new moment of inertia?

Answer 4

I new = I initial + mr²

Where mr² is the moment of inertia for
the small mass added.

Question 5

What affects the moment of inertia of a rotating object?

Answer 5

The mass and mass distribution of the
object, the position of the axis of rotation
and how far away the mass is from it.

Question 6

What is the rotational kinetic energy of an object?

Answer 6

Ek = ½ I⍵²

I is the moment of inertia (kg m2) and ω
is the angular speed (rad s-1).

Question 7

What is Torque?

Answer 7

A measure of how much a force causes
an object to rotate about an axis.
Measured in Newton meters (N m).

Question 8

What is a flywheel?

Answer 8

Flywheels are heavy wheels with a high
moment of inertia and angular momentum that convert inputted torque to rotational kinetic energy. They are used to store energy in a machine to be used at another point in a system.

Question 9

What factors affect the energy storage capacity of a flywheel?

Answer 9

1. Mass of the flywheel - As its mass increases, its moment of inertia will increase. As rotational kinetic energy is directly proportional to the moment of inertia, this will also increase meaning more energy can be stored by the flywheel.
2. Angular speed of the flywheel - As rotational kinetic energy is proportional to the square of the angular speed, if the flywheel’s angular speed increases, its rotational kinetic energy increases meaning more energy can be stored.
3. Friction - Even though they are very efficient, a flywheel will lose some of the energy that it stores to air resistance and friction between the wheel and its bearings, the effect of friction can be reduced by the following ways:
   ➔ Lubricating the bearings.
   ➔ Use bearings made of superconductors which would allow the flywheel to levitate.
   and have no contact with the bearing.
   ➔ Use flywheels in vacuums or sealed containers to reduce air resistance.
4. Using a flywheel that is spoked or has more of its mass concentrated at its edges will increase the amount of energy stored. This is because the moment of inertia of the object will be larger if most of its point masses are far from the axis of rotation.

Question 10

How are flywheels used for storing energy in braking vehicles?

Answer 10

When a vehicle brakes, normally its kinetic energy is wasted as heat in the brakes. Instead, a flywheel can be engaged to absorb that energy.
This is typically done by linking the wheels to the flywheel (directly or through a transmission). This converts the vehicle’s forward energy into rotational energy in the flywheel (making it spin faster).

The faster the flywheel spins, the more energy it stores. This energy is kept in rotation until it’s needed — for instance, when the vehicle starts accelerate again.

When the vehicle accelerates, the flywheel’s rotation is used to aid its forward movement. This happens by transferring the flywheel’s energy back to the wheels, reducing the load on the engine.

Question 11

How are flywheels used for smoothing torque and angular velocity?

Answer 11

Flywheels use each burst of power to charge and then smoothly deliver the energy to rotating components. For systems that exert a varying force
when the load torque is too high, the flywheel decelerates to top up the system and vice versa when engine torque is higher than load torque.

Question 12

How are flywheels used in production processes?

Answer 12

Most production, such as piercing sheets
of metal, require constant uniform action.
Attaching a flywheel to the motor
reduces problems by smoothing out
fluctuations.

Question 13

What are the advantages of flywheels?

Answer 13

● Very efficient.
● Long working life.
● Short recharge time.
● Environmentally friendly.

Question 14

What are the disadvantages of flywheels?

Answer 14

● Larger and heavier than other storage methods.
● Safety risk as wheel could break at high speeds.
● Energy lost through friction.
● Can oppose changes in direction for moving vehicles (but also helps improve stability).

Question 15

What is angular displacement?

Answer 15

The angle through which an object on a
circular path moves through. Measured
in radians.

Question 16

What are angular velocity and angular speed?

Answer 16

Angular speed = change in angle / time
(magnitude only). Measured in rad s-1

Angular velocity = change in angular
displacement / time (magnitude and
direction). Measured in rad s-1

Question 17

What is the relation between angular acceleration, angular velocity and angular displacement?

Answer 17

Angular displacement (θ) - the angle turned through in any given direction in radians.

Angular speed (ω) - the angle an object moves through per unit time (has only magnitude).
Units are rads-1
.
Angular velocity (ω) - the angle an object moves through per unit time (has magnitude and
direction, which can be either clockwise or anticlockwise). Units are rads-1
.
Angular acceleration (α) - the change in angular velocity over time taken. Units are rads-2

Question 18

Plot graphs of angular displacement against time for uniform and non-uniform acceleration.

Answer 18

For constant angular acceleration, angular displacement ∝ t2 shown by a smooth curve through the origin.

Question 19

Draw the graph for angular velocity against time when angular acceleration is constant.

Answer 19

Question 20

Draw the graph for angular velocity against time when angular acceleration is not constant

Answer 20

Question 21

Match up the linear quantities with their rotational equivalent.
s =
v =
a =

Answer 21

s = θ, angle through which a point has been rotated.

v = ⍵, angle a point rotates through per second.

a = 𝛼, rate of change of angular velocity.

Question 22

What are the uniform acceleration formulae (suvat) in terms of uniform angular acceleration?

Answer 22

Question 23

What equations can be used to calculate torque?

Answer 23

T = Fr = Force x perpendicular distance from axis of rotation to point of applied force

T = I𝛼 = Moment of Inertia x angular
acceleration.

Question 24

What is angular momentum?

Answer 24

I⍵ = moment of inertia x angular velocity.
Measured in N m s

Question 25

When is angular momentum conserved?

Answer 25

It is conserved only when no external forces act on the object.

Question 26

What is Angular Impulse?

Answer 26

The change in angular momentum or equivalently a product of a constant torque and time duration of that change. TΔt = Δ(I⍵)

Question 27

How does angular impulse relate to torque graphically?

Answer 27

Angular impulse is the area under the torque-time graph. Similar to Impulse = area under force-time graph.

Question 28

Work done (W)

Answer 28

is defined as the force causing a motion multiplied by the distance travelled. Work must be done on an object in order to make it rotate, therefore to calculate work done on a rotating object you must find the product of the torque and angular displacement. Similarly to linear dynamics, an increase in work done can increase the rotational kinetic energy of an object, however it must first overcome the frictional torque that may be present.

Question 29

How can you calculate the work done when rotating an object?

Answer 29

W = Tθ Work done = Torque x Angular displacement

Question 30

How can the work done be calculated graphically with a non-constant torque?

Answer 30

The work done is the area under the torque-angular displacement graph.

Question 31

What is the effect of frictional torque?

Answer 31

frictional torque is minimised in order to minimise the energy losses due to kinetic energy being transferred to heat and sound energy, however in some cases frictional torque can be useful. For example, when using a screwdriver you apply a frictional torque which increases the rotational kinetic energy.

Question 32

Power (P)

Answer 32

is the rate of energy transfer and as work is a measure of energy transfer, it is also the rate of doing work.

Question 33

How can you calculate the power expended in rotating an object?

Answer 33

P = T⍵ Power = Torque x Angular velocity

Question 34

Is there a frictional torque when rotating an object?

Answer 34

Yes, there is a frictional force resisting motion which causes a frictional torque, and some work will need to be expended in overcoming the frictional torque.

Question 35

The angular acceleration can be increased by:

Answer 35

● Increasing m and so increasing the torque ● Using a lighter wheel, which will decrease the moment of inertia of the wheel

Question 36

Define Angular momentum

Answer 36

is the product of the moment of inertia and angular velocity of an object, and its units are Nms.

Question 37

Angular momentum equation?

Answer 37

Angular momentum = Iω

Question 38

The law of conservation of angular momentum states

Answer 38

that when no external torque acts, the angular momentum of a system remains constant. This can be demonstrated by looking at the example of an ice skater spinning about a vertical axis on the tip of her skate, with her arms extended. As she moved her arms into her chest, she will begin moving faster - her angular velocity increases. This occurs because by moving her arms inwards she has decreased her moment of inertia, however angular momentum is conserved (as no external torque acts) so her angular velocity must increase.

Question 39

Due to the law of conservation of momentum, momentum before = momentum after. so give an equation

Answer 39

Question 40

Angular impulse

Answer 40

is the product of torque and its duration where the applied torque is constant, and is equal to the change in angular momentum. Angular impulse = change in angular momentum T Δt = Δ(Iω)

Question 41

Angular impulse can be found on a graph how?

Answer 41

found by calculating the area beneath a torque-time graph