1. Torque & Angular Acceleration AH Physics
Rotational Dynamics
Torque & Angular Acceleration
AH Physics

2. Torque
A force that produces a turning effect on an object is called a torque.
The turning effect depends on:
Size of the force (F) acting perpendicular to the rotation.
Distance from the axis of rotation to the force (r)
T = torque in newton metres (Nm)
F = force in newtons (N)
r = radius in metres (m)

3. Example
A driver is removing the wheel nuts from a car wheel when changing the tyre after getting a puncture. She applies force of 65 N to the wheel wrench (spanner) at a distance of 0.4m
Calculate the applied torque.
One of the wheel nuts is stuck. Suggest 2 ways in which she could apply a greater torque to remove the nut.
T = Fr 65 x 0.4 = 26 Nm.
Greater force at same distance or same force at greater distance (longer spanner)

4. Unbalanced torque Similar to Newtons 2nd Law F= ma
An unbalanced torque will produce an angular acceleration about an axis of rotation.
The rotational equivalent of mass is called the moment of inertia (I) of an object.
It depends on the mass of the object itself and the distribution of mass about its axis of rotation.
Moment of inertia is sometimes referred to as “rotational mass” and is very important in rotational motion.
T = torque measured in newton metres (Nm)
I = moment of inertia measured in kgm2
α = angular acceleration (rad s-2)

5. Moment of Inertia (I)
The moment of inertia of an object (I) is a measure of its reluctance to change its rotational motion. It depends on the mass of the object and the distribution of its mass about its axis of rotation.
The expression for the moment of inertia of a point mass (m) at a distance (r) from the axis of rotation can be derived by thinking about Newtons 2nd Law
Multiply both sides by r
Torque T = Fr
angular acceleration α =a/r
a= α r
Radius2 (m2)
Moment of inertia (kgm2)
Mass (kg)

6. Rotational kinetic energy
𝐸 𝑘 = 1 2 𝑚 𝑣 2
In linear motion, kinetic energy is by:
(All particles in an object moving at a constant linear velocity are moving at the same speed.)
In rotational motion, however, the particles in a disc rotating at a constant rate all have different speeds, a particle at the edge is moving faster than a particle near the axis of rotation, but……… they all have the same angular velocity
So we need a new relationship for kinetic energy that uses angular velocity.
The quantity (mr2) is called the moment of inertia (or “rotational mass”) of a point mass undergoing rotational motion. It is given the symbol I.

7. Moment of inertia for different objects
Different objects have different expressions for their moment of inertia. It depends on the distribution of the mass around the axis of rotation. Typical examples are given on the relationships sheet. (A capital M is often used to denote the total mass of the object.)
Think : Why is the moment of inertia of a disc half that of a hoop of the same mass?

8. Torque & Moment of inertia example (2017)
I = ½ mr2
I = ½ x 0.4 x 0.292
I = kgm2

9. 0.06 Nm.
3.5 rad s-2
15 rad s-1

10. Rotational Kinetic Energy
Rotational kinetic energy is similar to linear kinetic energy but you must use the moment of inertia (in place of mass) and the angular velocity
A washing machine drum can be considered to be a hollow cylinder (a hoop) of mass 8kg and radius 20cm. Determine the rotational kinetic energy of the drum when it is spinning at
a) 400 rpm b) 1600 rpm.
Moment of inertia of a hoop is (mr2). Angular velocity ω =2π/60 x rpm
a) 281 J b)4490 J

11. Total kinetic energy
If an object has a combination of linear and rotational motion e.g. a car wheel rolling along a road, you can determine the total kinetic energy simply by adding the linear (translational) and rotational kinetic energies.
This has some interesting consequences for cylinders rolling down a hill……

12. Rolling Cylinders - Demo
2 cylinders of the same mass and diameter are released from the same height on a slope. One is solid, the other is hollow.
Which one, if any, reaches the bottom first?
Think….
Does mass matter?
Does diameter matter?
Does length matter?
Walter Lewin demo
watch from 4:50 seconds.

13. Experiment – hollow and solid cylinders
Measure the mass of each cylinder.
Measure the radius of each cylinder
Calculate the moment of inertia of each cylinder.
By using conservation of energy, determine experimentally the moment of inertia of each cylinder:
Measure the vertical height of the slope.
Measure the length of the slope.
Measure the time taken for each cylinder to roll down the slope.
Determine the final linear and angular velocity of each cylinder at the bottom of the slope using
Determine experimentally the moment of inertia of each cylinder from these results using:
Which method is better? (Q paper)

14. Connected systems – The Flywheel (only for the brave)
A 2.50 kg mass is attached to a 25.0 kg flywheel with a long string that is wound onto the flywheel that has a radius of 0.30m and is free to rotate. The hanging mass is held stationary and then allowed to fall through a height of 2.00 m. As it falls the flywheel accelerates angularly.
Calculate the initial potential energy of the hanging mass.
Calculate the moment of inertia of the flywheel. (a disc)
Using conservation of energy, determine the final velocity of the mass.
Calculate the acceleration of the falling mass
Calculate the angular acceleration of the flywheel
49 J
1.125 kgm2
2.56 ms-1
1.64 ms-2
5.47 rad s-2

15. Total angular momentum before = total angular momentum after
Remember at Higher: momentum = mass x velocity
Well, at Advanced Higher we need to be able to understand angular momentum. Angular momentum = moment of inertia x angular velocity
Units kgm2 rads-1
Also:
The same rules apply :
“ angular momentum is conserved in the absence of any external torques”
Total angular momentum before = total angular momentum after
Angular momentum = linear momentum x radius

16. The ice skater
Conservation of angular momentum is neatly demonstrated by an ice skater or gymnast.
Arms out, large moment of inertia, small angular velocity
Bring your arms in: moment of inertia is reduced , therefore angular velocity increases!
Try it on a spinning chair with a couple of weights in your hands.

17. Example
An ice skater rotates at 3 rads-1 with her arms outstretched. Her moment of inertia in this position is 3.8 kgm2. She now bring her arms in and her moment of inertia decreases to 1.14 kgm2
a) Calculate her angular velocity when she brings her arms in.
b) Show that her rotational kinetic energy increases by 40J
c) Explain where this extra energy comes from.
𝑎) 𝐼 1 𝜔 1 = 𝐼 2 𝜔 ×3 = 𝜔 2
ω2 = 10 rad s-1
b) Ekb = 17 J Eka = 57 J Increase = 40 J.
c) Work done in pulling her arms in

18. Summary of Linear & Angular quantities
Linear quantity
symbol
Angular quantity
Relationship
Displacement s
Angular displacement
velocity v
Angular velocity
acceleration a
Angular acceleration
Kinetic energy Ek
Rotational Kinetic Energy
momentum p
Angular momentum
force F
Torque

19. Summary of Linear & Angular quantities
Linear quantity
symbol
Angular quantity
Relationship
displacement s
angular displacement θ
velocity v
angular velocity ω
acceleration a
angular acceleration α
kinetic energy Ek
rotational kinetic energy
momentum p
angular momentum L
force F
torque T