ROTATIONAL MOTION
This topic involves things spinning around their ………………………………………………….
Examples:
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
Measuring Rotational Motion:
Angular displacement
Measuring rotational motion can be tricky, because each point
travels a different distance at different speeds.
Rather than talk about displacement we talk about
……………………………displacement.
This is simply the angle the object rotates through.
This is the same for any part of the object.
Angular displacement can be measured in revolutions, degrees or more commonly
radians.
R
Angle in radians is:  =

d
If
d=R
 =………… radian
 =3600,
If
One revolution = 3600 = ………… radians
d = …………
1800 = ………… radians
Angular Speed
Recall:
linear speed is
v
d
t
angular speed is

=
ex1. Calculate the angular speed of a merry go round doing 6.0 revolutions in 30s.
ex2. Through what angle will it turn in 5.0 minutes?
What does the linear speed “v” depend on?
V

v
d r


t
t
This shows the relationship between linear speed and
angular speed.
Angular Acceleration
This is when the angular speed ……………………………… or …………………………………
Recall: linear acceleration = a = ______
Likewise angular acceleration =
=
as above, (v = r ), so linear acceleration = a =
ex1. A merry go round starts from rest. After 20s it is spinning at 0.10rev/s.
a) Calculate the new angular velocity in rad/s
b) Calculate the angular acceleration
c) Calculate the linear speed and acceleration of a person standing 3.0m from
the centre.
ex2. A cycle with 70cm diameter wheels is rolling at 3.5 ms-1
a) Calculate the angular speed of the wheels.
b) How many times do they turn in 1.0 second?
3.5ms-1.
c) Extension: What is the linear speed of the top and bottom of the wheel
Extension
Combined Motion
Sketch the path of a piece of chewing gum on the rim of a bike wheel and a train wheel.
front
view
side
view
side
view
front
view
Homework exercises:
1) Calculate the angular velocity and linear velocity of a person at the equator.
(R=6300km)
Explain the difference between angular velocity and linear velocity.
2) A CD spins at 3000 rpm. Calculate the angular velocity in rad s-1.
Calculate the linear velocity of a point on the rim, and a point half way to the
centre.
It takes 3.0 s to reach this speed, calculate the angular acceleration.
What causes Angular Acceleration?
To make something accelerate you need a ………………………… .
Likewise, to produce an angular acceleration requires a ……………………………… .
Remember from last year (ha ha) that Torque is the ……………………………………………………………
It depends on:
The size of the …………………………
the ………………………………………………………………………
Equation …………………………
Example
Max exerts a …………………………………
torque about the pivot.
Min exerts a ………………………………….
torque about the pivot.
Min
Max
3.0m
2.0m
The see saw is balanced so the two ……………………… are …………………… and ……………………………
Min is 75kg. Calculate Max’s mass.
Homework exercise.
Describe what is meant by:
Angular displacement
Angular velocity
Angular acceleration
Demonstration. Predict which torque produces the greatest angular
acceleration.
Prediction:
………………………………………………………………………………………
Result:
………………………………………………………………………………………
Explanation:
So we can say that the bigger the torque, the bigger the …………………………………………
or ……………………………………………… is proportional to …………………………………………
In Newton’s Second Law,
acceleration 
force
mass
What is the rotational equivalent of mass or inertia?
Rotational Inertia
We know that to change an object’s linear velocity requires a …………………… .
Likewise to change an object’s angular velocity requires a ………………………… .
Mass, or ………………………… means “how hard it is to ……………………………… an object”.
The rotational equivalent of mass is ……………………………… …………………… .
It means “how hard it is to ……………………………… an object”.
i.e. A bicycle wheel is much ……………………
to spin than a car wheel because the bicycle
wheel has ………………………… rotational inertia.
i.e. A ferris wheel is much ……………………
to spin than a fan because the ferris wheel
has ………………………… rotational inertia.
Compare translation and rotation:
Acceleration = __________
Angular acceleration = __________
Equation:
An object’s inertia (or ………………………… ) only depends on the amount of “stuff” in it.
What do you think rotational inertia depend on?
……………………………………………………………………………………………………………………………………………
……………………………………………………………………………………………………………………………………………
(1) Rotational Inertia and Mass
Polystyrene
Wood
Same
radius
and
torque
Predict which wheel will have the greatest angular acceleration
Prediction:
………………………………………………………………………………………………………………………………………………………………
Observation
………………………………………………………………………………………………………………………………………………………………
The string applies a …………………… to the wheel, causing …………………… ……………………………
The wooden wheel has …………………… angular acceleration than the polystyrene, because it
has …………………………… rotational inertia. (This means it is …………………………… to spin)
I  ……………………
Rotational inertia depends on ……………………
(2) Rotational Inertia and Mass Distribution
Both wheels have the
same mass and the
same Torque acting.
Predict which wheel has the greatest angular acceleration.
Prediction
………………………………………………………………………………………………………………………………………………………………
Observation
………………………………………………………………………………………………………………………………………………………………
The string applies a …………………… to the wheel, causing ………………………… …………………………
The large radius wheel has …………………… angular acceleration than the small radius one,
because it has …………………………… rotational inertia.
Rotational inertia depends on ……………………
………………………
I  ……………………
( “r”is the ……………………………… ………………………)
Homework exercise
What does rotational inertia mean?
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
Explain which has more rotational inertia a large 250 g metal can or a small 250g metal
can.
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
Extension: Explain in terms of energy why a larger wheel has more rotational inertia
than a smaller wheel of the same mass.
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
Time keeping
This clock has a timing mechanism called
a Torsional Pendulum. The brass disc
oscillates about a vertical axis through
the centre. Explain the purpose of the
two weights
Homework exercise:
Explain what is meant by:
Torque.
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
Rotational Inertia
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
A similar example:
Compare the two angular accelerations.
Prediction:
………………………………………………………………………………………………………………………………………………………………
Explanation
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
As you probably guessed, the rotor on the left is ………………………………… to spin. It has a
………………………………… angular acceleration because its mass is distributed
………………………………… from the centre of rotation. (i.e. it has …………………………………
rotational inertia.)
The “hands on method”. Hold the two tubes (equal mass) in each hand and try to spin
them about their central axis. Explain what you notice.
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
So for translation,
and for rotation
acceleration=
force
mass
angular acceleration =
torque
rotationl inertia
or
In summary we can say that rotational inertia depends on two things:
1) ………………………………………………………………………….
2) ………………………………………………………………………….
Rotational Inertia for this ring is:
R
I=
n.b. This equation is only used when all the mass is the same distance from the centre.
e.g a ……………………………………………………………………
Where the mass is evenly distributed, (e.g ………………………………………………………)
The rotational inertia will be …………………… because the average mass is
…………………… to the centre than it is for a ring.
(We say the effective radius is ………………………………… )
For a uniform solid disc:
I=
ex. Use dimensional analysis to check the equation for the rotational inertia.
Rotational Inertia = I =
Torque
ang. acceln
=
Homework Exercise
1) Explain why a ring is harder to spin than a solid disc (of the same mass and radius)
………………………………………………………………………….………………………………………………………………………….…………
……………………………………………………………….………………………………………………………………………….……………………
…………………………………………………….………………………………………………………………………….………………………………
2) You have two hoops of the same mass. One has twice the radius. Explain why the
larger one needs twice the force at the rim to cause the same angular acceleration
………………………………………………………………………….………………………………………………………………………….…………
……………………………………………………………….………………………………………………………………………….……………………
…………………………………………………….………………………………………………………………………….………………………………
The Great Can Race
The cans have the same mass and the same torque acting. Explain why one can is easier
to accelerate.
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
A solid disc races a cylinder. Explain which one gets to the bottom first.
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
Homework exercise
1) (excellence) A grindstone has an 11cm radius and a mass of 4.0 kg. It is spinning at
2500 rads-1 when it is turned off. It takes 2.0 minutes to stop. Calculate the frictional
torque.
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
2)
(excellence) A 120 kg satellite is shaped like a solid drum with a radius of 0.50 m.
It starts from rest and after 1 minute spins once every second. One small rocket on
the rim produces the force. Calculate minimum force.
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
3).
A ferris wheel (FW) has a mass of 750kg (including passengers) and a radius of
4.0m. A small wheel applies a friction force of 130N to it.
a) Calculate the torque applied to the ferris wheel.
b) Calculate the rotational inertia. (State any
assumptions)
c) Calculate the angular acceleration.
4) Explain how these answers will change if the passengers moved closer to the
centre.
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
Extension. Derive an equation for the angular acceleration of
the wheel in terms of the Tension, wheel’s mass (M) and
wheel’s Radius (R).
Harder Extension. Apply F = ma to the falling mass (m).
Derive an equation for the falling mass’s linear acceleration
in terms of m, M, R and g
Homework exercises.
Hannah has a bike wheel with a mass of 250 g and a radius of
74 cm.
1) Estimate the rotational inertia. What assumptions are you
making?
2) She applies a 5.0 N force with her finger. Calculate the
angular acceleration.
3) She wants to maximise her acceleration in a sprint race. Is it more important to
reduce the mass of the wheel’s hub or the rim? Explain
Angular Momentum
Linear momentum is a measure of how hard it is to ……………. something that is moving.
Linear Momentum = …………………………………… x ……………………………………
The rotational equivalent of linear momentum is …………………………………… momentum. It is a
measure of how hard it is to …………………………………… something that is spinning.
Angular Momentum = ……………………… ……………………… x ………………… …………………
L = …………………
(…………………)
Clearly a big wheel spinning fast is …………………………… to stop than a big wheel spinning
slow, and a big wheel spinning is …………………………… to stop than a small wheel spinning at
the same speed .
This idea can be extended to find the angular momentum of a point mass. (this is jargon
for a small lump of stuff)
L=I
L=
I = mr2
 = v/r
nb. this is the angular momentum of a point mass about
some point “r” away.
Extension: A 220g tennis ball spins in a 3.0m circle at 20ms-1.
Calculate the angular momentum about:
x
y
Remember:
L=
x
y
To change an object’s momentum requires a ……………………………………
To change an object’s angular momentum requires a ……………………………………
This leads to the idea of…
Conservation of Angular Momentum
“If the net torque is zero, the total angular momentum is ……………………………………”
i.e. L
= I
is constant if
 = ………
Demonstrations of Conservation of Angular Momentum.
1)
A ball orbits around in a circle. Predict what will
happen if the radius is decreased.
…………………………………………………………………………………………………
…………………………………………………..……………………………………………
Describe what does happen and explain what happens.
…………………………………………………………………………………………………
…………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
2)
As Rosie pulls her arms and legs in, her rotational inertia …………………………… (her
effective radius …………………………)
Her angular momentum stays ………………………… ( because the torque on her is
……………………)
So angular velocity must …………………………………… .
i.e
Li = Lf
I1 1 = …………………
So, if rotational inertia decreases, angular velocity …………………………………
3)
Extension
Explain what happens when Rachel inverts the wheel.
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
4). Set a cell phone on vibrate, and place it on a smooth surface. Call the phone. Use
conservation of angular momentum to explain what is inside it, and why it rotates.
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
Homework exercises:
1)
Explain the difference between rotational inertia and angular momentum.
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
2)
Explain what is meant by conservation of angular momentum.
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
3)
A spinning bike wheel slows down. Why does this not break the law of
conservation of angular momentum?
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
4) Divers and trampolinists try to make as many turns as possible.
i.e. They try to maximise their ………………………… ……………………………… .
Once they are spinning, they ……………………………. This means they
………………………………………………………………
This reduces their ……………………………………………………………… and increases their
…………………………………………..
Helicopters.
The rotor gains angular momentum in the …………………………………… direction. (from above).
To stop the body spinning in the opposite direction, a ………………………… is needed. This is
provided by the ……………… ………………… . It applies a ……………………………….. torque.
Which way does it push?
………………………………………………………………………………………………………………………………………………………………
Extension: One problem for astronauts in orbiting space stations is changing the space
stations orientation. . Explain how a fixed electric motor with a heavy wheel could be
used to cause it to spin and change its orientation.
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
ex1. A 250g turntable with a 15cm radius spins once every 2.5s.
An identical turntable is dropped onto it.
(i) What quantity is conserved?
(ii) Calculate the new angular velocity.
(iii) Repeat (ii) for a dropped turntable with the same mass but half the radius.
ex2. A 100kg woman runs a 10ms-1 tangential to the rim of a stationary merry go round.
(i) Estimate the rotational inertia of the merry go round. (m= 1000kg r = 3.0m)
Discuss your assumptions.
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
(ii) Calculate the angular momentum of the woman about the centre of the merry go
round.
(iii) Assuming she gives all her angular momentum to the MGR, calculate the angular
velocity of the MGR after she lands on it.
(Explain if it is true that she gives all her angular momentum to the MGR.)
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
Extension: calculate the actual angular velocity.
(iv) Explain what happens to the angular velocity if she then walks toward the centre.
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
Rotational Kinetic Energy
Energy is the ability to do …………………………
Give examples of how a spinning object can do work:
………………………………………………………………………………………………………………………………………………………………
As the weight falls, it loses ……………………………………………………
What happens to this energy?
(remember Total Energy is always …………………………)
………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………
……………………………………………………………………
Rotational Energy Equation:
Recall KE = ………………………… Have a guess at the equation for rotational kinetic energy.
RKE =
Lets check: Consider a spinning ring.
Kinetic Energy =
=
1
1
2
mv 2
2
m(
)2 .
( v =r)
=
Rotational KE =
ex1. Consider the previous example of a weight on a string attached to a wheel.
Write an equation for the energy change:
Extension
Write an equation for the velocity of the falling weight in terms of the height it falls.
Why is the acceleration of the falling weight not 9.8 ms-2?
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
Consider the turntable drop example. Is it an elastic collision? Explain
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
Demonstrations
Consider a cart racing a wheel down a slope. Predict which will win.
………………………………………………………………………………………………………………………………………………………………
Observe and explain which wins.
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
………………………………………………………………………………………………………………………………………………………………
……………………
ex1. Tash slows a spinning turntable (solid disc) by holding her finger against the edge.
(radius = 20 cm. Tangential force = 2.0 N )
The turntable goes from 5.0 rad/s to rest in 10 s.
a. Calculate the turntable’s rotational inertia.
b. calculate the turntable’s mass.
c. Calculate the work done by her finger.