Rotational dynamics

Question 1

What is inertia?

Answer 1

The resistance to a change of motion in linear motion
The larger the mass of an object, the greater the inertia

Question 2

What is the moment of inertia?

Answer 2

The rotational equivilant of inertia for linear motion
The resistance to a change of rotational motion, depending on the distribution of mass around a chosen axis of rotation

Question 3

What does the moment of inertia measure?

Answer 3

How ‘easy’ or ‘hard’ it is to rotate an object

Question 4

How do you calculate moment of inertia when a system has more than one component rotating the same axis?

Answer 4

We calculate the moment of inertia of each individual component seperately, using their individual mass and radius, and then add it together for the overall moment of inertia

Question 5

How do you calculate moment of inertia?

Answer 5

Moment of inertia = mass x radius^2

Question 6

Why do we need moment of inertia?

Answer 6

The moment of inertia in rotational dynamics plays the same roll as mass in linear motion
Instead of using mass in equations, we use moment of inertia

Question 7

What is rotational kinetic energy?

Answer 7

A body moving with linear motion has an associated linear kinetic energy. Similarily, a rotating object has an associated rotational kinetic energy

Question 8

How do you calculate rotational kinetic energy?

Answer 8

It is given by its moment of inertia (in place for mass) and angular velocity (in place for velocity)
Rotational kinetic energy = 1/2 x moment of inertia x angular velocity^2

Question 9

What is rolling motion?

Answer 9

Circular objects are made to move with both linear and rotational motion
Rolling motion is a combination of rotating and sliding (tranational) motion

Question 10

How does angular and linear velocity differ as a disc rotates?

Answer 10

Each point on the disc has a different angular velocity depending on its distance from the centre
The linear velocity is the same at all points on the circumference

Question 11

What happens when a disc rolls without sliping?

Answer 11

There is enough friction present to initiate rotational motion
The point in contact with the surface has a velocity of 0
The centre of mass has a velocity of angular velocity x radius
The top point has a velocity of angular velocity x radius^2

Question 12

What happens to GPE when rolling down a slope?

Answer 12

As the object rolls down the slope, the gravitational potential energy is transferred to both the linear (translational) kinetic energy and rotational kinetic energy as 2 seperate stores

Question 13

What is angular displacement?

Answer 13

The change in angle (radians) through which a rigid body has rotated relative from point A to point B

Question 14

What is angular velocity?

Answer 14

The rate of change of angular displacement, with respect to time

Question 15

What is angular acceleration?

Answer 15

The rate of change of angular velocity, with respect to time

Question 16

How do you calculate angular displacement, angular velocity and angular acceleration?

Answer 16

AD: θ (radians)
AV: ω = θ / ∆t
AA: α = ∆ω / ∆t

Question 17

Angular equivalents for linear variables

Answer 17

s = θ (displacement)
u = ωi (initial velocity)
v = ωf (final velocity)
a = α (acceleration)
t = t (time)

Question 18

Uniform angular acceleration equations

Answer 18

The linear acceleration equations (suvat) can be swapped out for their angular equivalents to get the angular acceleration equations
ωf = ωi + αt

Question 19

What is torque?

Answer 19

Torque is the change in rotational motion due to a turning force
Torque = applied force x perpendicular distance from axis
T (Nm) = F (N) x r (m)

Question 20

What is the torque of a couple?

Answer 20

The sum of the moments produced by each of the forces in the moment

Question 21

Why is there no net force of a couple?

Answer 21

The forces in a couple are equal and opposite hence do not produce a resultant force
Due to Newton’s second law, no resultant force means a couple won’t cause an object to accelerate, it will only rotate with a constant angular velocity

Question 22

Newton’s second law in rotational dynamics

Answer 22

The torque required to give a rotating object a certain angular acceleration is calculated using T = Iα (F=ma)

Question 23

What is the difference between T = Iα and T = Fr

Answer 23

T = Fr : how much torque a force produces
T = Iα : how that torque changes the rotation of the object

Question 24

How do you calculate angular momentum?

Answer 24

Angular momentum = moment of inertia x angular velocity
L = Iω

Question 25

Conservation of angular momentum

Answer 25

The angular momentum of a system always remains constant unless a net torque is acting on the system An increase in radius results in an increase in moment of inertia hence, a decrease in angular velocity

Question 26

What is angular impulse?

Answer 26

An average resultant torque acting for a certain amount of time Change in angular momentum

Question 27

How do you calculate angular impulse?

Answer 27

∆L = T∆t = ∆(Iω)

Question 28

What is rotational work done and how do you calculate it?

Answer 28

Work must be done on an object to make it rotate a certain distance Work done (J) = torque (Nm) x angular displacement (rad/s) W = Tθ

Question 29

How does work done relate to kinetic energy?

Answer 29

Work done can also be calculated by change in rotational kinetic energy Final rotational kinetic energy - initial rotational kinetic energy

Question 30

What is frictional torque?

Answer 30

The difference between the applied torque and the resulting net torque

Question 31

How do you calculate frictional torque?

Answer 31

Frictional force x radius Moment of inertia x average deceleration

Question 32

Why do we minimise frictional torque?

Answer 32

To minimise the kinetic energy that is lost to heat and sound

Question 33

What is power?

Answer 33

Rate of energy transfer Rate of doing work

Question 34

How do we calculate rotational power?

Answer 34

Power (watts) = torque (Nm) x angular valocity (rad/s) P = W / t = Tθ / t = Tω

Question 35

What is the purpose of flywheels?

Answer 35

Flywheels are used in machines to act as an energy resvoir - to store and supply energy when it is required They consist of a heavy metal disc or wheel that has a large moment of inertia - high mass and large radius

Question 36

How does a flywheel work?

Answer 36

As a flywheel spins, its input torque is converted into rotational kinetic energy which is stored in the flywheel. This is a result of resisting the changes to rotation. The greater the moment of inertia, the greater the resistance and hence, the greater the energy stored.

Question 37

Uses of flywheels

Answer 37

Smoothening torque and speed Regnerative braking Production processes

Question 38

Why is a wheel preferred to a disc in flywheels?

Answer 38

Wheels have a greater moment of inertia compared to a disc of the same mass. Therefore, they will have a greater resistance to changes in rotation and will therfore hold more energy

Question 39

How do flywheels help in smoothening torque and speed?

Answer 39

Power in an engine is not produced continuoysly, but rather only in the 'power stroke' or 'combustion' part of the energy cycle, so it is released in bursts. This causes the engine to produce a torque that fluctates. The torque makes the flywheel rotate, moving the vehicle forwards If the torque is uneven, it will cause a jerky motion. This is a waste of energy and unomfortable for passangers The flywheel added will speed up or slow down over a period of time because of its inertia and will therefore smoothen the sharp fluctations in torque The greater the moment of inertia of the flywheel, the smaller the fluctations in speed

Question 40

How do cars use flywheels in regenerative braking?

Answer 40

In conventional braking, the kinetic energy store of the vehicle is transferred as waste through to the thermal energy store Instead, when regenerative braking is applied, the flywheel is engaged and will "charge up" using the energy lost by braking When the vehicle needs to accelerate later, the energy stored by the flywheel is used to do this

Question 41

How are flywheels used in production processes?

Answer 41

An electric motor in industrial machines can be used alongside a flywheel The motor is used to charge up the flywheel, which can then transfer short bursts of energy This prevents the motor from stalling and a less powerful motor can be used

Question 42

What factors affect the energy storage capacity of a flywheel?

Answer 42

Mass of the flywheel Angular speed of the flywheel Friction Shape of the flywheel

Question 43

How does mass affect the energy storage capacity of a flywheel?

Answer 43

Moment of inertia is directly proportional to mass, as mass increases, so does moment of inertia. The rotational kinetic energy is directly proportional to moment of inertia. Hence, increase in mass results in increase of rotational kinetic energy storage.

Question 44

How does angular speed affect the energy storage capacity of a flywheel?

Answer 44

The rotational kinetic energy is proportional to the square of the angular speed

Question 45

How does friction affect the energy storage capacity of a flywheel?

Answer 45

Flywheels can lose energy to friction and air resistance between the wheels and bearings. It can be reduced byb using lubrication between bearings, using bearings made of semiconductors (so the wheels can levitate) or by using the flywheel in a vacuum (to minimise air resistance)