1. Rotational Kinematics
Lecture 14
Rotational Kinematics

2. Office Hours:
Several requests to meet today
I’ll be available 1:30 – 2:30

3. Rolling Motion
If a round object rolls without slipping, there is a fixed relationship between the translational and rotational speeds:

4. Rolling Motion
We may also consider rolling motion to be a combination of pure rotational and pure translational motion: = +

5. Rolling Wheel
A wheel rolls without slipping. Which vector best represents the velocity of point A? a) b) c) A
Answer: b
e) the velocity at point A is zero d)

6. + Rolling Wheel A v vrot vtrans
A wheel rolls without slipping. Which vector best represents the velocity of point A? a) b) c) A
Answer: b
e) the velocity at point A is zero d) + v
vrot
vtrans

7. Rotational Kinetic Energy
For this mass,

8. Rotational Kinetic Energy
For these two masses,
Ktotal = K1 + K2 = mr2ω2

9. Moment of Inertia We can also write the kinetic energy as
Where I, the moment of inertia, is given by
What is the moment of inertia for two equal masses on the ends of a (massless) rod, spinning about the center of the rod?

10. Moment of Inertia We can also write the kinetic energy as
Where I, the moment of inertia, is given by
What is the moment of inertia for a uniform ring of mass M and radius R, rolling around the center of the ring?

11. Moment of Inertia for various shapes
Moments of inertia of various objects can be calculated:
This is a concept and will not be on the formula sheet!
and so one can calculate the kinetic energy for rotational motion:

12. A block of mass 1.5 kg is attached to a string that is wrapped around the circumference of a wheel of radius 30 cm and mass 5.0 kg, with uniform mass density. Initially the mass and wheel are at rest, but then the mass is allowed to fall.
What is the velocity of the mass after it falls 1 meter?

13. A block of mass 1.5 kg is attached to a string that is wrapped around the circumference of a wheel of radius 30 cm and mass 5.0 kg, with uniform mass density. Initially the mass and wheel are at rest, but then the mass is allowed to fall.
What is the velocity of the mass after it falls 1 meter?

14. Energy of a rolling object
The total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies:
Since velocity and angular velocity are related for rolling objects, the kinetic energy of a rolling object is a multiple of the kinetic energy of translation.

15. Conservation of Energy
An object rolls down a ramp - what is its translation and rotational kinetic energy at the bottom?

16. Conservation of Energy
An object rolls down a ramp - what is its translation and rotational kinetic energy at the bottom?
From conservation of energy:
Note: no dependence on mass, only on distribution of mass
Velocity at any height:

17. Conservation of Energy
If these two objects, of the same radius, are released simultaneously, which will reach the bottom first?
The disk will reach the bottom first – it has a smaller moment of inertia. More of its gravitational potential energy becomes translational kinetic energy, and less rotational.

18. Conservation of Energy
If these two objects, of the same radius, are released simultaneously, which will reach the bottom first?
What if they have different radii?
mR2 cancels, leaving only the geometric factor (in this case, 1/2 vs. 1)

19. Rolling Down
Two spheres start rolling down a ramp from the same height at the same time. One is made of solid gold, and the other of solid aluminum.
Which one reaches the bottom first?
a) solid aluminum
b) solid gold
c) same
d) can’t tell without more information
Answer: b

20. Mass and radius don’t matter, only the distribution of mass (shape)!
Rolling Down
Two spheres start rolling down a ramp from the same height at the same time. One is made of solid gold, and the other of solid aluminum.
Which one reaches the bottom first?
a) solid aluminum
b) solid gold
c) same
d) can’t tell without more information
Answer: b
Moment of inertia depends on mass and distance from axis squared. For a sphere:
I = 2/5 MR2
But you don’t need to know that! All you need to know is that it depends on MR2
initial PE: mgh
final KE:
MR2 cancels out!
Mass and radius don’t matter, only the distribution of mass (shape)!

21. Moment of Inertia
Two spheres start rolling down a ramp at the same time. One is made of solid aluminum, and the other is made from a hollow shell of gold.
Which one reaches the bottom first?
a) solid aluminum
b) hollow gold
c) same
d) can’t tell without more information

22. Moment of Inertia
Two spheres start rolling down a ramp at the same time. One is made of solid aluminum, and the other is made from a hollow shell of gold.
Which one reaches the bottom first?
a) solid aluminum
b) hollow gold
c) same
d) can’t tell without more information
A shell has a larger moment of inertia than a solid object of the same mass, radius and shape
initial PE: mgh
final KE:
A solid sphere has more of its mass close to the center. A shell has all of its mass at a large radius. Total mass and radius cancel in expression for fraction of K tied up in angular (rolling) motion.
Larger moment of inertia -> lower velocity for the same energy.

23. ConcepTest at a greater height as when it was released.
A ball is released from rest on a no-slip surface, as shown. After reaching its lowest point, the ball begins to rise again, this time on a frictionless surface as shown in the figure. When the ball reaches its maximum height on the frictionless surface, it is:
at a greater height as when it was released.
at a lesser height as when it was released.
at the same height as when it was released.
impossible to tell without knowing the mass of the ball.
impossible to tell without knowing the radius of the ball.

24. Q: What if both sides of the half-pipe were no-slip?
ConcepTest
A ball is released from rest on a no-slip surface, as shown. After reaching its lowest point, the ball begins to rise again, this time on a frictionless surface as shown in the figure. When the ball reaches its maximum height on the frictionless surface, it is:
at a greater height as when it was released.
at a lesser height as when it was released.
at the same height as when it was released.
impossible to tell without knowing the mass of the ball.
impossible to tell without knowing the radius of the ball.
Q: What if both sides of the half-pipe were no-slip?

25. The pulsar in the Crab nebula was created by a supernova explosion that was observed on Earth in a.d Its current period of rotation (33.0 ms) is observed to be increasing by x 10-5 s per year.
What is the angular acceleration of the pulsar in rad/s2 ?
Assuming the angular acceleration of the pulsar to be constant, how many years will it take for the pulsar to slow to a stop?
Under the same assumption, what was the period of the pulsar when it was created?

26. The pulsar in the Crab nebula was created by a supernova explosion that was observed on Earth in a.d Its current period of rotation (33.0 ms) is observed to be increasing by x 10-5 s per year.
What is the angular acceleration of the pulsar in rad/s2 ?
Assuming the angular acceleration of the pulsar to be constant, how many years will it take for the pulsar to slow to a stop?
Under the same assumption, what was the period of the pulsar when it was created?

27. The pulsar in the Crab nebula was created by a supernova explosion that was observed on Earth in a.d Its current period of rotation (33.0 ms) is observed to be increasing by x 10-5 s per year.
What is the angular acceleration of the pulsar in rad/s2 ?
Assuming the angular acceleration of the pulsar to be constant, how many years will it take for the pulsar to slow to a stop?
Under the same assumption, what was the period of the pulsar when it was created?
(a)

28. The pulsar in the Crab nebula was created by a supernova explosion that was observed on Earth in a.d Its current period of rotation (33.0 ms) is observed to be increasing by x 10-5 s per year.
What is the angular acceleration of the pulsar in rad/s2 ?
Assuming the angular acceleration of the pulsar to be constant, how many years will it take for the pulsar to slow to a stop?
Under the same assumption, what was the period of the pulsar when it was created?
(b)
(c)

29. Power output of the Crab pulsar
Power output of the Crab pulsar, in radio and X-rays, is about 6 x 1031 W (which is about 150,000 times the power output of our sun). Since the pulsar is out of nuclear fuel, where does all this energy come from ?
The angular speed of the pulsar, and so the rotational kinetic energy, is going down over time. This kinetic energy is converted into the energy coming out of that star.
calculate the change in rotational kinetic energy from the beginning to the end of a second, by taking the moment of inertia to be 1.2x1038 kg-m2 and the initial angular speed to be 190 s-1. Δω over one second is given by the angular acceleration.

30. Power output of the Crab pulsar
Power output of the Crab pulsar, in radio and X-rays, is about 6 x 1031 W (which is about 150,000 times the power output of our sun). Since the pulsar is out of nuclear fuel, where does all this energy come from ?
The angular speed of the pulsar, and so the rotational kinetic energy, is going down over time. This kinetic energy is converted into the energy coming out of that star.
calculate the change in rotational kinetic energy from the beginning to the end of a second, by taking the moment of inertia to be 1.2x1038 kg-m2 and the initial angular speed to be 190 s-1. Δω over one second is given by the angular acceleration.

31. Lecture 14
Rotational Dynamics

32. Moment of Inertia The moment of inertia I:
The total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies:

33. Torque
We know that the same force will be much more effective at rotating an object such as a nut or a door if our hand is not too close to the axis.
This is why we have long- handled wrenches, and why doorknobs are not next to hinges.

34. The torque increases as the force increases, and also as the distance increases.

35. Only the tangential component of force causes a torque

36. A more general definition of torque:
Fsinθ
Fcosθ
You can think of this as either:
- the projection of force on to the tangential direction OR
- the perpendicular distance from the axis of rotation to line of the force

37. Torque
If the torque causes a counterclockwise angular acceleration, it is positive; if it causes a clockwise angular acceleration, it is negative.

38. e) all are equally effective
Using a Wrench a c d b
You are using a wrench to tighten a rusty nut. Which arrangement will be the most effective in tightening the nut?
e) all are equally effective

39. e) all are equally effective
Using a Wrench a c d b
You are using a wrench to tighten a rusty nut. Which arrangement will be the most effective in tightening the nut?
Because the forces are all the same, the only difference is the lever arm. The arrangement with the largest lever arm (case b) will provide the largest torque.
e) all are equally effective

40. The gardening tool shown is used to pull weeds. If a 1
The gardening tool shown is used to pull weeds. If a 1.23 N-m torque is required to pull a given weed, what force did the weed exert on the tool?
What force was used on the tool?

41. Force and Angular Acceleration
Consider a mass m rotating around an axis a distance r away.
Newton’s second law:
a = r α
Or equivalently,

42. Torque and Angular Acceleration
Once again, we have analogies between linear and angular motion:

43. The L-shaped object shown below consists of three masses connected by light rods. What torque must be applied to this object to give it an angular acceleration of 1.2 rad/s2 if it is rotated about
(a) the x axis,
(b) the y axis
(c) the z axis (through the origin and perpendicular to the page)
(a)
(b)
(c)

44. The L-shaped object shown below consists of three masses connected by light rods. What torque must be applied to this object to give it an angular acceleration of 1.2 rad/s2 if it is rotated about
an axis parallel to the y axis, and through the 2.5kg mass?

45. Dumbbell I
a) case (a)
b) case (b)
c) no difference
d) it depends on the rotational inertia of the dumbbell
A force is applied to a dumbbell for a certain period of time, first as in (a) and then as in (b). In which case does the dumbbell acquire the greater center-of-mass speed ?
Answer: c

46. Dumbbell I
a) case (a)
b) case (b)
c) no difference
d) it depends on the rotational inertia of the dumbbell
A force is applied to a dumbbell for a certain period of time, first as in (a) and then as in (b). In which case does the dumbbell acquire the greater center-of-mass speed ?
Because the same force acts for the same time interval in both cases, the change in momentum must be the same, thus the CM velocity must be the same.