Engineering - Rotational Dynamics

Question 1

Angular velocity eq?
Units?

Answer 1

ω = Δθ/Δt in rads^-1
(rate of change of angular displacement)

Question 2

1 revolution is equal to 2π radians.
How to get from rpm to rads^-1 ?

Answer 2

rpm x 2π / 60 = rads^-1

Question 3

Angular acceleration, symbol α, arises when a particle moving in a circular path..?

Answer 3

increases or decreases its speed of rotation.

Question 4

Equation for angular acceleration?

Answer 4

α = Δω/Δt

Question 5

An increase in angular velocity increases tangential velocity ∴
(+eq)

Answer 5

∴ increase in tangential acceleration too.
a = rα

Question 6

Define torque, T?
Eq? Units?

Answer 6

T = product of a (rotational vs linear in mechanics moments) force and the perpendicular distance of the line of action of the force from a particular point or axis of rotation.
T = Fs (Nm)

Question 7

A positive torque produces a —- rotation.

Answer 7

clockwise

Question 8

T = Fs and
a = rα ∴

Answer 8

T = mr^2α

(linear F = ma)

Question 9

Define inertia.

Answer 9

The resistance of a physical object to a change in its state of motion.

Question 10

Define moment of inertia, I ?

Answer 10

measure of the extent to which the object resists being ROTATIONALLY accelerated about a certain axis.

Question 11

Moment of inertia eq?
Units?

Answer 11

For a particle of mass m, that is a distance r from an axis of rotation, the moment of inertia I = mr^2
Units kgm^2.

(I = Σmr^2 ∴ position of axis of rotation affects value for I)

Question 12

What is a flywheel?

Answer 12

A flywheel is a disc of wheel that can rotate rapidly. As it spins, it stores rotational Ek which can be released and used subsequently.

Question 13

The greater the — of a rotating object, the more Ek it can store.

Answer 13

moment of inertia

(Much like the greater the mass the greater the translational Ek (=1/2mv^2).)

Question 14

When two rotating objects eg two flywheels share a common axis of rotation, then their individual moments of inertia..?

Answer 14

are added together to give a moment of inertia for the combines system.
ΣT = Σ (mr^2α) = Σ(mr^2)α = Iα
(since α same for whole object)
This is newton’s second law for rotational motion.

Question 15

Ek of rotating object eq?

Relate to flywheel.

Answer 15

Ek = 1/2m (rω)^2 = 1/2(mr^2)ω^2 = 1/2Iω^2

∴ I and ω need to be as large as possible to maximise energy storage capacity of flywheel.

Question 16

PAG: moment of inertia of a large flywheel can be determined by using a hanging mass to produce a torque on the flywheel.
Give eq.

Answer 16

mgΔh = 1/2mv^2 + 1/2Iω^2

Question 17

In industry and in transport applications, flywheels have 3 main uses:

Answer 17

1) they can SMOOTH OUT TORQUE OR SPEED variations in vehicles.
2) they can recover and RE-USE some of the Ek that would otherwise be wasted, through braking for example.
3) they’re an essential COMPONENT OF MACHINERY in some production processes.

Question 18

1) Smoothing torque and speed:
-In engines power is not produced continuously ∴ engine produces a torque that ….

Answer 18

fluctuates!

Question 19

1) Smoothing torque and speed:
-Torque is what makes the wheel rotate thereby moving the vehicle forwards. Uneven torque —-> —>

Answer 19

jerky motion + unwanted vibrations —> uncomfortable and a waste of energy.

Question 20

1) Smoothing torque and speed:
- Flywheel added will speed up and slow down …… —>

Answer 20

speed up and slow down over a period of time bc of its inertia —> sharp fluctuations in torque are flattened/smoothed.

Question 21

1) Smoothing torque and speed:
- In practice, to increase pressure and further control fluctuations, real engines have …

Answer 21

4+ combustion cylinders. These are staggered in their operation and are all attached collectively to a large flywheel, which smooth/average out the torques being produced by the separate cylinders.

increasing number of cylinders = more smooth but also= increased cost, weight and complexity

Question 22

2) Kinetic energy recovery system (KERS):
- uses … braking where energy from a car’s braking motion is ‘collected’ and stored for re-use.

Answer 22

regenerative braking

Question 23

2) Kinetic energy recovery system (KERS):
- eg in hybrid cars, there’s a combination of conventional combustion engines and an electric motor. This involves generators (convert mechanical E to electrical E), which …

Answer 23

are spun as braking occurs. This produces electricity, which is stored in batteries. When needed, these batteries power an electric motor (the generator working backwards) to contribute to the driving torque of the conventional engine.

Question 24

2) Kinetic energy recovery system (KERS):
-However, a drawback is that the batteries have a large …

Answer 24

mass, so the overall fuel efficiency isn’t as good as might be expected.

Question 25

2) Kinetic energy recovery system (KERS):
-Flybrid (flywheel + hybrid) system is when vehicle brakes, some of the Ek is used to spin a flywheel. This can be achieved in two ways: either through a generator/motor method similar to hybrid, or by a direct mechanical linkage via gears between the rear driveshaft and the flywheel.
When accelerating…

Answer 25

-When accelerating, reverse situation: flywheel rotational energy is transferred back to the driveshaft as required, done by direct mechanical method or by producing electricity which is then used to spin motors attached to the wheel of the car. Control of flow of power to and from flywheel is computerised.

Question 26

3) Production application:
Many industrial manufacturing applications eg piercing sheet metal, rely on continuous, non-fluctuating action. Traditionally, leather belt, driven by electric motors, were used to drive presses. Flywheel essential bc belts could ….
Belts replaced with….

Answer 26

stretch or warp w diff atm conditions, or could become inflexible or slip momentarily.
Later, belts replaced by motors directly attached to presses. But motors can have non-regularities, so solution is to have motors spin large flywheels. Can also add flywheel to conveyor belt to smooth sudden stops.

Question 27

Why is a flywheel fitted between the motor and the stamping machine? (3 marks)

Answer 27

• flywheel acts as a store of Ek.
• able to deliver large amounts of Ek in short time.
• w/o flywheel, motor would STALL during stamping (as load torque excessively high).

Question 28

Angular momentum:
Angular momentum, L, of a particle about an axis =
(in words and symbols)

Answer 28

the product of its linear momentum and the perpendicular distance of the particle from the axis.
L = mvr = mωr^2 = Iω (=T/α ω)

Question 29

Angular momentum has direction which is along the axis of rotation. By convention, the direction of the angular momentum vector is ….

Answer 29

towards an observer if the direction of the rotation is anticlockwise

(kinda like the right hand rule in magnetic fields)

Question 30

What’s the principle of conservation of angular momentum?

Answer 30

The total angular momentum of a system at some initial time is equal to the total angular momentum at some later time, provided no external torque acts upon the system.

Question 31

Ice skater brings arms closer to body - explain this situation:

Link to angular momentum.
Link this to Ek?

Answer 31

Mass closer to axis of rotation —> lower moment of inertia. Angular momentum (Iω) conserved ∴ decrease in I increases ω ∴ skater spins faster.
There’s a corresponding increase in rotational Ek (=1/2Iω^2). This energy has come from the skater, who had to do work to pull her arms in.

Question 32

Gyroscopes:

First things first - watch the video!

Answer 32

Ok done :)

https://www.bing.com/videos/search?q=veritasium+gyroscopic+precession+youtube&view=detail&mid=1390A906A5774F2B97F71390A906A5774F2B97F7&FORM=VIRE

Question 33

Gyroscopes:
Define gyroscope.

Answer 33

A device consisting of a wheel or disc that spins rapidly about an axis that is also free to change direction.

Question 34

Gyroscope:
-As a consequence of the conservation of angular momentum, if it’s spun pointing in a certain direction eg vertically then as long as it keeps spinning quite fast…

Answer 34

it’ll continue to point that same way. Orientation of axis unaffected by tilting or rotating the mounting.

Question 35

Gyroscope:
-Friction will cause gyroscope to lose energy –> slows down –> axis will move slightly away from its initial position (due to friction w surface, air pressure, non-uniform mass distribution etc). As soon as this starts to occur there’ll be … This causes a….

Answer 35

an external torque due to gravity. This causes a change in angular momentum.

Question 36

Gyroscope:
-The result is a continuously changing direction of axis - the gyroscope, as well as spinning on its axis, rotates slowly around the vertical. This is an effect called ?

Answer 36

Precession! The direction of precession will be in the same sense (cw or acw) as the spinning of the gyroscope itself.

Question 37

Angular impulse:
If …. there’ll be a change of angular momentum, called angular impulse. Eq?

Answer 37

If an external torque is applied..
ΔL = I(ω2-ω1)

Question 38

Relate impulse to torque?

Answer 38

T = Iα = I Δω/Δt = ΔL/Δt (torque is rate of change of angular momentum)
∴ ΔL = TΔt
∴ Δ(Iω) = TΔt

Question 39

Work and power in rotating systems:
Work done is a product of the force and the distance moved. Eq?

Answer 39

W = Frθ = Tθ

θ in radians

Question 40

Work done may increase rotational Ek of rigid body, but must also overcome resistive forces. Minimise by?

Answer 40

using lubrication or ball bearing.

Question 41

The frictional force in the case of an object rotating on an axis - like a flywheel on an axle - is at the edge of the rotating object, so there’ll be a …. about the axis, called….

Answer 41

torque about the axis, called frictional torque.

Question 42

A frictional torque may act to resist the motion of a rotating body, and to reduce its …

Answer 42

rotational Ek.

Question 43

Alternatively, a frictional torque may be Applied - such as when you grip and turn a screwdriver - in which case the rotational Ek will ..
Whenever a torque turns an object through an angle…

Answer 43

Increase.
Whenever a torque turns an object through an angle, work is done.

Question 44

Power is the rate of doing work. Give eq

Answer 44

P = ΔW/Δt = Δ(Tθ)/Δt = T Δθ/Δt = Tω