1. Rolling, Torque, and Angular Momentum
Circular Motion II
Rolling, Torque, and Angular Momentum

2. Torque, Energy and Rolling
Torque, moment of inertia
Newton 2nd law in rotation
Rotational Work
Rotational Kinetic Energy
Rotational Energy Conservation
Rolling Motion of a Rigid Object
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3. Mechanical Energy Conservation Review
When Wnc = 0,
The total mechanical energy is conserved and remains the same at all times
Remember, this is for conservative forces, no dissipative forces such as friction can be present
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4. Preview: Total Energy of a System
A ball is rolling down a ramp
Described by three types of energy
Gravitational potential energy
Translational kinetic energy
New: Rotational kinetic energy
Total energy of a system
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5. Energy in Rotational Motion
A rotating rigid body consists of mass in motion, so it has Kinetic Energy
Recall: m2 m3 r2 r3
Therefore:
Now:
Let
This is called the Moment of Inertia
This is the Rotational Kinetic Energy Equation of a rigid body
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6. Example A C B
mA=2.0 kg
mB=3.0 kg
mC=1.0 kg
3.0 m
5.0 m
4.0 m
What is the Moment of Inertia of this object when it is rotated through Point A, perpendicular to the plane of the Diagram?
What is the Moment of Inertia about the axis coinciding with axis BC?
If the Object rotates about Point A, perpendicular to the plane with angular speed of 5.0 rad/s, what is its Kinetic Energy?
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7. The Moment of Inertia about Point A axis is 84 kgm2
Solution A C B
mA=2.0 kg
mB=3.0 kg
mC=1.0 kg
3.0 m
5.0 m
4.0 m
What is the Moment of Inertia of this object when it is rotated through Point A, perpendicular to the plane of the Diagram?
Since the Mass at Point A lies on the axis of Rotation, its distance r from the origin is zero, so it contributes nothing to the Moment of Inertia
December 29, 2018
The Moment of Inertia about Point A axis is 84 kgm2

8. The Moment of Inertia about BC axis is 18 kgm2
Solution A C B
mA=2.0 kg
mB=3.0 kg
mC=1.0 kg
3.0 m
5.0 m
4.0 m
What is the Moment of Inertia about the axis coinciding with axis BC?
Since the Masses at Point B and C lies on the axis of Rotation, its distance r from the origin is zero, so they contributes nothing to the Moment of Inertia
December 29, 2018
The Moment of Inertia about BC axis is 18 kgm2

9. Solution A C B
mA=2.0 kg
mB=3.0 kg
mC=1.0 kg
3.0 m
5.0 m
4.0 m
If the Object rotates about Point A, perpendicular to the plane with angular speed of 5.0 rad/s, what is its Kinetic Energy?
Since the Mass at Point A lies on the axis of Rotation, its distance r from the origin is zero, so it contributes nothing to the Moment of Inertia
December 29, 2018

10. Definition of Torque First we need definition of torque:
Direction: right hand rule.
Torque is calculated with respect to (about) a point (Fulcrum). Changing the point can change the torque’s magnitude and direction
Magnitude:
A torque is an influence which tends to change the rotational motion of an object. One way to quantify a torque is
Torque = Force applied x lever arm
The lever arm (r┴) is defined as the perpendicular distance from the axis of rotation to the line of action of the force.
Fingers will point along r, ensure they will bend in direction of F, thumb will point along axis of rotation in the direction of the Torque.

11. Torque The units of torque are force times distance, or newton-meters.
A torque of 1 N-m (not Joules) is created by a force of 1 newton acting with a lever arm of 1 meter.
Example:
You exert a Force on a door of 50N, applied perpendicular to the plane of the door. The door is 1.0m wide. Assuming that you pushed the door at its edge, what was the torque on the swinging door (taking the hinge as the pivot point)?
Via the Right Hand Rule, the direction of the Torque is out of the screen (positive, counterclockwise)
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12. These torques are clockwise about A, so negative
Example
A truck is on a bridge. Given that the bridge weighs 80000kg and the distance between spans A and B is 50.0m. The truck, whose mass is kg, is located 15.0m from span A. Determine the support forces at both ends of the bridge (use 10 m/s2 for g).
In torque problems with heavy extended objects, just assume that the object’s weight is hanging at the centre of mass
Torque at B
This positive torque is counter clockwise, because it is pushing upward on the bridge pivoted at A
These torques are clockwise about A, so negative

13. Example
A truck is on a bridge. Given that the bridge weighs 80000kg and the distance between spans A and B is 50.0m. The truck, whose mass is kg, is located 15.0m from span A. Determine the support forces at both ends of the bridge.
Force at B
Force at A
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14. Torque and angular Acceleration
Recall:
This is Newton’s Second Law for Rotation
Only the net torque can cause an angular acceleration
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15. Only tangential momentum component contribute
Angular Momentum I
Every rotational quantity that we have encountered is the analog of some quantity in translational motion.
The analog of Momentum of a particle is Angular Momentum
The vector quantity of Angular Momentum is denoted by:
r is the particle’s instantaneous position vector
p is its instantaneous linear momentum
The Magnitude of Angular Momentum is denoted by:
l Is the perpendicular distance from velocity (momentum) of particle to origin
December 29, 2018
Only tangential momentum component contribute

16. Angular Momentum II
Same basic techniques that were used in linear motion can be applied to rotational motion.
F becomes 
m becomes I
a becomes 
v becomes ω
x becomes θ
Linear momentum defined as
What if mass of center of object is not moving, but it is rotating?
Angular momentum

17. Angular Momentum II Angular momentum of a rotating rigid object
L has the same direction as 
L is positive when object rotates in CCW
L is negative when object rotates in CW
Angular momentum SI unit: kgm2/s
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18. Example
Calculate the Angular Momentum of a 10 kg mass rotating 9 cm about an origin, given that the angular velocity is 320 radians per second
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19. Rotational Inertial of Rigid Object
You do not have to memorize these, as the Moment of Inertial for a Specific Rigid Object will be given to you.
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20. Example
Calculate the Angular Momentum of a 10 kg rigid disk rotating 9 cm about its axis, given that the angular velocity is 320 radians per second, and the Moment of Inertial of a disk is 1/2mr2.
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21. What Rolls Down Faster
Two solid cylinders roll, without slipping, down an incline, Their initial gravitational potential energy is being converted into two types of kinetic energy: translational KE and rotational KE. One cylinder is 10 times heavier than the other, who wins the race.
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22. Hey, mass and radius play no part in the final velocity
Hey, mass and radius play no part in the final velocity. They both get to the bottom at the same time.
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23. Angular momentum III Angular momentum of a system of particles
angular momenta add as vectors
be careful of sign of each angular momentum
for this case:
Right Hand Rule demonstrate that L1 is out of screen and L2 is into screen
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24. Example
Two objects are moving as shown in the figure . What is their total angular momentum about point O? m2 m1
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25. Angular Momentum Conservation
where i denotes initial state, f is final state
L is conserved separately for x, y, z direction
For an isolated system consisting of particles,
For an isolated system is deformable
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26. Angular Momentum Conservation
Angular Momentum is conserved any time an object, or system of objects experiences no net torque
Isolated system: net external torque acting on a system is ZERO
no external forces
net external force acting on a system is ZERO
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27. Change I by curling up or stretching out - spin rate w must adjust
Controlling spin (w) by changing I (moment of inertia):
In the air, tnet = 0
L is constant
Change I by curling up or stretching out
- spin rate w must adjust

28. Example
A skater is spinning (making one revolution every 2.0 seconds) with her arms extended and with 5.0 kg in each hand. She pulls her hands to her stomach. If the original distance the weights are from her axis is 1.0 m and when her arms are tucked in this distance is 0.20 m. If her moment of inertia without the weights is 3.0 kgm2 when arms are stretched out and 2.2 kgm2 when arms are in tight, determine her final angular velocity.
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29. Solution The conservation of angular momentum must hold
The moment of Inertia of the system is:
Therefore:
Inertial angular velocity of the system is:
The Final moment of Inertia is:
The conservation of angular momentum must hold
Therefore:
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30. Work done by a pure rotation
Apply force F to mass at point r, causing rotation-only about axis
Find the work done by F applied to the object at P as it rotates through an infinitesimal distance ds
Only transverse component of F does work – the same component that contributes to torque
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31. Work-Kinetic Theorem pure rotation
As object rotates from i to f , work done by the torque
I is constant for rigid object
Power
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32. Work-Energy Theorem For pure translation For pure rotation
Rolling: pure rotation + pure translation
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33. Example
An motor attached to a grindstone exerts a constant torque of 10 Nm. The moment of inertia of the grindstone is I = 2 kgm2. The system starts from rest.
Find the kinetic energy after 8 s
Find the work done by the motor during this time
Find the average power delivered by the motor
Find the instantaneous power at t = 8 s
December 29, 2018

34. We have I, we need to find ω
Solution
An motor attached to a grindstone exerts a constant torque of 10 Nm. The moment of inertia of the grindstone is I = 2 kgm2. The system starts from rest.
a) Find the kinetic energy after 8 s
We have I, we need to find ω
December 29, 2018

35. There are two ways to Solve this problem
Solution
An motor attached to a grindstone exerts a constant torque of 10 Nm. The moment of inertia of the grindstone is I = 2 kgm2. The system starts from rest.
b) Find the work done by the motor during this time
There are two ways to Solve this problem
Logic
Algebra
It was the motor that gave the Grindstone all of the kinetic energy. So, the work done by the motor is 1600J

36. Solution
An motor attached to a grindstone exerts a constant torque of 10 Nm. The moment of inertia of the grindstone is I = 2 kgm2. The system starts from rest.
c) Find the average power delivered by the motor
d) Find the instantaneous power at t = 8 s
December 29, 2018

37. Energy Conservation (Deeper Understanding)
When Wnc = 0,
The total mechanical energy is conserved and remains the same at all times
Remember, this is for conservative forces, no dissipative forces such as friction can be present
December 29, 2018

38. Total Energy of a Rolling System
A ball is rolling down a ramp
Described by three types of energy
Gravitational potential energy
Translational kinetic energy
Rotational kinetic energy
Total energy of a system
December 29, 2018

39. A Ball Rolling Down an Incline Example
A ball of mass M and radius R starts from rest at a height of h and rolls down a 30 slope, what is the linear speed of the ball when it leaves the incline? Assume that the ball rolls without slipping.

40. Rotational Work and Energy Understanding
A ball rolls without slipping down incline A, starting from rest. At the same time, a box starts from rest and slides down incline B, which is identical to incline A except that it is frictionless. Which arrives at the bottom first?
Ball rolling:
Box sliding:
Therefore since the box’s final velocity is faster, it will arrive first.

41. Example: A Non-isolated System
A sphere mass m1 and a block of mass m2 are connected by a light cord that passes over a pulley. The radius of the pulley is R, and the mass of the thin rim is M. The spokes of the pulley have negligible mass. The block slides on a frictionless, horizontal surface. Find an expression for the linear acceleration of the two objects.
December 29, 2018

42. same result followed from earlier method using 3 FBD’s & 2nd law
Masses are connected by a light cord
Find the linear acceleration a.
Use angular momentum approach
No friction between m2 and table
Treat block, pulley and sphere as a non- isolated system rotating about pulley axis. As sphere falls, pulley rotates, block slides
Constraints: I
Ignore internal forces, consider external forces only
Net external torque on system:
Angular momentum of system:
(not constant)
same result followed from earlier
method using 3 FBD’s & 2nd law
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43. Example
A puck of mass m = 0.5 kg is attached to a taut cord passing through a small hole in a frictionless, horizontal surface. The puck is initially orbiting with speed vi = 2 m/s in a circle of radius ri = 0.2 m. The cord is then slowly pulled from below, decreasing the radius of the circle to r = 0.1 m.
What is the puck’s speed at the smaller radius?
Find the tension in the cord at the smaller radius.
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44. Solution a) Given and Required:
m = 0.5 kg, vi = 2 m/s, ri = 0.2 m, rf = 0.1 m, vf = ?
Isolated system?

45. Solution b) Given and Required:
m = 0.5 kg, vi = 2 m/s, ri = 0.2 m, rf = 0.1 m, vf = 4 m/s
December 29, 2018

46. Example: A merry-go-round problem
A 40-kg child running at 4.0 m/s jumps tangentially onto a stationary circular merry-go-round platform whose radius is 2.0 m and whose moment of inertia is 20 kg-m2. There is no friction.
Find the angular velocity of the platform after the child has jumped on.
December 29, 2018

47. The Merry-Go-Round
The moment of inertia of the system = the moment of inertia of the platform plus the moment of inertia of the person.
Assume the person can be treated as a particle
As the person moves toward the center of the rotating platform the moment of inertia decreases:
The angular speed must increase since the angular momentum is constant:
December 29, 2018

48. Conservation of angular Momentum
Solution: A merry-go-round problem
Conservation of angular Momentum
Iplatform = 20 kg.m2
VT = 4.0 m/s
mchild = 40 kg
r = 2.0 m
wplatform = 0 rad/s
Therefore:
December 29, 2018

49. Example
A 2m bar is held horizontally and is used to open a crate. The tip of the bar acts as the fulcrum and is inserted 4 cm between the lid and the crate. If 15 N of force is applied perpendicularly to the end of the bar. What is the force applied to the crate lid?
A) 345 N
B) 453 N
C) 508 N
D) 642 N
E) 750 N
0.04m 2m
15N
December 29, 2018

50. Example
A large crane picks up a 60 m steel beam. The steel beam is uniform, but the crane cable is attached 3.0 m off centre of the beam. How much vertical force must be applied on the guide rope to keep the beam level given that the beam has a mass 4.0 kg/m and the guide rope is place on the shorter end of the beam ?
A) 261 N
B) 453 N
C) 508 N
D) 642 N
E) 735 N
27m 3m
60m
The guide rope is 27m from the fulcrum, and the centre of mass is 3m from the fulcrum
December 29, 2018

51. Example
A 3m long bar is held horizontally with the fulcrum positioned 1 metre from the right end. How much vertical force must be applied to the left end of the bar to support a 800 N block on the right end?
A) 345 N
B) 453 N
C) 500 N
D) 348 N
E) 400 N 1m 3m
December 29, 2018

52. Example
Newton’s First Law states that a particle is in equilibrium if the net forces acting on it are zero. When do you consider torques in Newton’s First Law?
A) When Forces are unbalanced
B) You never consider Torques in Newton’s First Law
C) When using objects that have dimensions
D) Torques are not forces so don’t include
E) Torques always cancel out so don’t include
When objects have dimensions, they may have forces that act off-axis which angularly accelerate the object, so must be considered
December 29, 2018

53. Example What happens to an object that is experiencing a net torque?
A) Nothing
B) The object accelerates linearly
C) The velocity of the object increases
D) The object accelerates angularily
E) The object accelerates both linearly and angularly
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54. Multi-Choice
When a cow is moving in a circular motion, what accelerations are NOT possible?
A) Zero Acceleration
B) Inward Acceleration
C) Tangential Acceleration
D) Angular Acceleration
E) Outward Acceleration
All objects moving in a circular motion must experience an acceleration towards the centre. If there is a net torque on the cow, it may also experience angular acceleration. If there is a net force tangent to the circular motion (its hooves), if could also experience tangential acceleration. That leaves the other two out of the picture.
December 29, 2018

55. Example
Four identical rods of length L experience a force as seen below. Rank the magnitude of the torque about the pivot point on the right end of the rod. F F 2. 1.
F/2 F F 2F 3.
60o 4.
1=3=4>2
December 29, 2018

56. Example
A ruler of length L rotates with a constant velocity, ω, measured in radians per second. Determine the Period of the ruler’s rotation.
A) 1/ω
B) Ω
C) 2π/ω
D) ω/2π
E) 2π
Period = 1/f, so need to convert angular velocity into angular frequency before inverting.
December 29, 2018