Here are the homework problems from Section 8.5 of the textbook: 81. A figure skater with arms extended has a moment of inertia of 2.5 kg·m2. She pulls her arms in so her moment of inertia decreases to 1.8 kg·m2. If initially she was rotating at 5.0 rad/s, what is her final angular speed? 83. A solid cylinder of mass 2 kg and radius 0.1 m rolls without slipping down an inclined plane that makes an angle of 30° with the horizontal. If its initial speed is 5 m/s, find its angular speed and linear speed when it reaches the bottom of the plane. 84. A solid sphere of mass 2

1. Physics - Chapter 7
Rotational Motion and Equilibrium

2. Physics - Chapter 7
Rotational Motion and Equilibrium
By
Professor AJAL A J
Head of Academy

3. Chapter 7: Rotational Motion and Equilibrium
7.1 Rigid Bodies, Translation, and Rotation
7.2 Torque, Equilibrium, and Stability
7.3 Rotational Dynamics
7.4 Rotational Work and Energy
7.5 Angular Momentum

4. Rigid Bodies, Translation, and
Rotation

5. Rigid Bodies, Translation, and Rotation
Please Do Now
How do you think the tight-rope walkers are
able to keep from falling?
Write three lines using full sentences.

7. 8.1 Rigid Bodies, Translation, and Rotation
rigid body – an object or system of particles in which the distances between
particles are fixed and remain constant.
Solids can be considered to be rigid bodies for the purpose of analyzing
rotational motion.
Fluids are not considered to be rigid bodies.
translational motion – every particle in the object has the same instantaneous
velocity. There is no rotation.
Write two examples.
rotational motion – motion about a fixed axis. All the particles of an object have
the same instantaneous angular velocity and travel in circles about the axis of
rotation. Write two examples.
rigid body motion – is usually a combination of translational and rotational
motion. Write two examples.

9. 7.2: Torque and Equilibrium

10. 7.2 Torque and Equilibrium
τ = r F sinϕ

11. 7.2 Torque and Equilibrium
•
The sign convention for torques is as follows:
-The torque, which rotates or tends to rotate the body clockwise (CW), is
negative.
-The torque, which rotates or tends to rotate the body counterclockwise
(CCW), is positive.
The SI unit of torque is the m·N (meter-newton).

12. 7.2 Torque and Equilibrium
The perpendicular distance r┴ from the axis of rotation to the line of
action of a force is called the moment arm (or lever arm) and is
equal to r sin θ.

13. 7.2 Torque and Equilibrium
The same force in the opposite direction with a smaller moment arm
produces a smaller torque in the opposite direction.

14. 7.2 Torque and Equilibrium
When a force acts through the axis of rotation, the torque is zero.

15. 8.2 Torque and Equilibrium
Think of a real-world example of torque.
Turn and talk to your shoulder partner.

16. 8.2 Torque and Equilibrium
equilibrium – a state in which things are in balance or are stable.
translational equilibrium – the sum of the forces on a body is zero
• the body remains at rest or at constant velocity
•  Fi = F1 + F2 + F3 + …. = 0
rotational equilibrium – the sum of torques on a body is zero
• the body is rotationally at rest or rotates with a constant angular velocity
•  τi = τ 1 + τ 2 + τ 3 + …. = 0
The board is in translational The board is in translational but not
and rotational equilibrium. rotational equilibrium.
F F
F
F

17. 8.2 Torque and Equilibrium
mechanical equilibrium – when the conditions for translational and rotational
equilibrium are satisfied.
 Fi = 0 (for translational equilibrium)
 τi = 0 (for rotational equilibrium)
static equilibrium – the condition that exists when a rigid body remains at rest
What are some examples of static translational equilibrium?
What are some examples of static rotational equilibrium?
Problem-Solving Hint: Use the convention counter-clockwise (ccw) is positive and
clockwise (cw) is negative.

18. 8.2 Torque and Equilibrium
Exit Ticket
Write at least (3) lines to answer the question:
How did Philippe Petit demonstrate torque,
translational and rotational equilibrium?
Turn in your ticket
when you are done.

20. 8.2 Torque and Equilibrium
Please Do Now
Answer the question:
What is torque? Describe it in your own words.

21. 8.2 Torque and Equilibrium
Example 8.1: The bolts on a car wheel require tightening to a torque of 90 m·N.
If a 20 cm long wrench is used, what is the magnitude of the force required
a) when the force is perpendicular to the wrench,
b) when the force is 35° to the wrench as shown.
c) Why does it need more force in (b) than in (a)?

22. 8.2 Torque and Equilibrium
Example 8.2: A uniform board of weight 40 N supports two children weighing 500 N
and 350 N, respectively. If the support is at the center of the board and the 500 N
child is 1.5 m from the center, what is the position of the 350 N child?

23. 8.2 Torque and Equilibrium
Example 8.3: A 10 m long uniform beam weighing 100 N is supported by two ropes
at the ends as shown. If a 400 N person sits at 2.0 m from one end of the beam,
what are the tensions in the ropes?

24. 8.2 Check for Understanding
1.

25. 8.2 Check for Understanding
1.

26. 8.2 Check for Understanding
2.

27. 8.2 Check for Understanding
2.

28. 8.2 Check for Understanding
(Hint: Use worker A as the pivot point.)
3.

29. 8.2 Check for Understanding
a3.

30. 8.2: Stability and Center of Gravity

31. 8.2 Stability and Center of Gravity
center of gravity (CG) – the point located at the object’s average position of
weight.
∙ for a symmetrical object, it is at the geometric center (ex: baseball)
∙ for an irregularly shaped object, the CG is at the heavier end (ex: baseball bat)
∙ for objects not made of the same material throughout, the CG is located at the
heaviest part
(ex: Weebles)
center of mass – the average position of all
particles of mass that make up an object.
∙ note: on earth, CG = CM

32. 8.2 Stability and Center of Gravity
Rule:
∙ If the CG of an object is above the area of support, the object will remain
upright.
∙ If the CG extends outside the area of support, the object will topple.
Topple!

33. 8.2 Stability and Center of Gravity
unstable equilibrium – any movement lowers its center of gravity.
stable equilibrium – any movement raises its center of gravity.
neutral equilibrium – movement does not cause a change in CG.

34. 8.3 Rotational Dynamics

35. Warmup: Torque Your Way In Physics Warmup # 71
An object rotates because a torque acts on it. When you exert a force at the end
of a wrench in order to rotate it (along with the bolt it is attached to), you may not
have thought about the fact that you applied the force perpendicular to the
handle. The longer the wrench’s handle, the less force you have to apply,
because the amount of torque is equal to the product of the force times the
distance to the axis about which it is rotating.
_________________________________________________________________
Use the relationship of torque to force and distance to explain the following two
small mysteries in your kitchen:
1. Why is it easier to open a cabinet door when the doorknob is at the end of the
door than when it is in the middle of the door?
2. Why is a big doorknob easier to turn than a small one?

36. .3 Rotational Dynamics

37. 8.3 Rotational Dynamics

38. Moments of
inertia for
some
uniform
objects

39. 8.3 Rotational Dynamics
Example 8.5: A solid cylinder of mass 10 kg is pivoted about a frictionless axis
through its center O. A rope wrapped around the outer radius R1 = 1.0 m, exerts a
force of F1 = 5.0 N to the right. A second rope wrapped around another section of
radius R2 = 0.50 m exerts a force of F2 = 6.0 N downward.
a) What is the angular acceleration of the disk?
b) If the disk starts from rest, how many radians does it rotate through in the first
5.0 s?
F1
F2
R1
R2
O

40. 8.3 Check for Understanding
1.

41. 8.3 Check for Understanding
1.

42. 8.3 Check for Understanding
2.

43. 8.3 Check for Understanding

44. 8.3 Check for Understanding
3.

45. 8.3 Check for Understanding
3.

46. 8.4 Rotational Work and Kinetic Energy

47. 8.4 Rotational Work an Kinetic Energy

48. 8.4 Rotational Work an Kinetic Energy

49. 8.4 Rotational Work an Kinetic Energy
Rotational Translational
Work: W =  W = Fd = Fx
Kinetic Energy: K = ½ I 2 K = ½ mv2
Power: P =  P = Fv
Work-Energy Theorem: W = K = ½ I 2 - ½ I o
2 W = ½ mv2 - ½ mvo
2
The kinetic energy of a rolling body (without slipping) relative to an axis through
the contact point is the sum of the rotational kinetic energy about an axis through the
center of mass and the translational kinetic energy of the center of mass.
K = ½ ICM 2 + ½ mvCM
2
total = rotational + translational
KE KE + KE

50. 8.4 Rotational Work an Kinetic Energy
Example 8.6: A solid sphere of mass 1.0 kg and radius 0.010 m rolls with a speed
of 10 m/s. How high up an inclined plane can it roll before it comes to rest?

51. 8.4 Rotational Work an Kinetic Energy
Example 8.7: A cylindrical hoop of mass 10 kg and radius 0.20 m is accelerated
by a motor from rest to an angular speed of 20 rad/s during a 0.40 s interval.
a. How much work is required?
b. What is the power output of the motor?

52. 8.4 Check for Understanding
1. From W = , the unit(s) of rotational work is
a. watt
b. Nm
c. kgrad/s2
d. Nrad
Answer: b

53. 8.4 Check for Understanding
2. A bowling ball rolls without slipping on a flat surface. The ball has
a. rotational kinetic energy
b. translational kinetic energy
c. both rotational and translational kinetic energies
Answer: c

54. 8.4 Check for Understanding
3.

55. 8.4 Check for Understanding

56. Homework

57. 8.5 Angular Momentum

58. 8.5 Angular Momentum
Newton’s 2nd law can be written as:
F = ma  F = m v or F =  p
 t  t
The rotational form of Newton’s 2nd law:
 = L
t
 = net torque
L = change in angular momentum
t = change in time
Translational Analogy: p = mv

59. 8.4 Angular Momentum
 The SI unit of angular momentum is kgm2/s.
 The vector direction of angular momentum is determined by the right hand
rule as applied to angular velocity.
Net Torque on a Rotating Rigid Body
 The net torque of a rotating rigid body is equal to the time rate of change of
angular momentum, L /t. In other words, a net torque causes a change in
angular momentum.
 The SI units of torque is mN.

60. 8.5 Angular Momentum

61. 8.5 Angular Momentum
Examples of Conservation of Angular Momentum:
- enormous wind speeds of hurricanes and tornados
- passing a football; spin of a bullet; gyroscope
- others:

62. 8.5 Angular Momentum
Example 8.8: A figure skater rotating at 4.00 rad/s with arms extended has a
moment of inertia of 2.25 kgm2. If she pulls her arms in so the moment of inertia
decreases to 1.80 kg m2, what is the magnitude of her final angular speed?

63. 8.5 Check for Understanding
1. The units of angular momentum are
a) Nm
b) kg m/s2
c) kg m2/s
d) Jm
Answer: c

64. 8.5 Check for Understanding
2.

65. 8.5 Check for Understanding
2.

66. 8.5 Check for Understanding
3. The release of vast amounts of carbon dioxide may increase the Earth’s
average temperature through the greenhouse effect and cause the polar ice caps
to melt. If this occurred and the ocean level rose substantially, what effect would it
have on the Earth’s rotation and on the length of the day?
Answer: The polar ice caps (with almost zero moment of inertia) will go to the
ocean and increase the moment of inertia of the Earth. This results in a slower
rotational speed or a longer day.

67. 8.5 Check for Understanding
4.

68. 8.5 Check for Understanding
4.

69. Homework 8.C (Section 8.5)
• HW 8.C: p.291-293: 81, 83 - 87, 92, 100, 101, 105.