1. Rotational Motion and Equilibrium
Ch. 8
Rotational Motion and Equilibrium

2. Ch. 8 Overview Rolling Motion Torque Rotational Equilibrium
Rotational Dynamics
Rotational Kinetic Energy
Conservation of Angular Momentum

3. Tangential Variables
To describe the connection between linear and rotational motion, we use tangential variables.
Arc length
Tangential speed
Tangential acceleration

4. Arc Length
Δs = rΔθ Δθ r Δs

5. Tangential Speed Δθ r Δs vt
but ω=Δθ/Δt so

6. Ex. The turntable turns at 33 1/3 rpm
Ex. The turntable turns at 33 1/3 rpm. Find the angular velocity of the motor if the radius of the small pulley is 1.27 cm.

7. Gear Ratio v is the same for both wheels r1ω1 = r2ω2 ω2 = r1/r2 ω1
γ = r1/r2 is called the gear ratio v r1 r2

8. ½ the ladybug’s twice the ladybug’s equal to the ladybug’s
A ladybug sits at the outer edge of a merry go-round, and a gentleman bug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second. The gentleman bug’s angular speed is
½ the ladybug’s
twice the ladybug’s
equal to the ladybug’s
cannot be determined 1 2 3 4 5

9. ½ the ladybug’s twice the ladybug’s equal to the ladybug’s
A ladybug sits at the outer edge of a merry go-round, and a gentleman bug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second. The gentleman bug’s tangential speed is
½ the ladybug’s
twice the ladybug’s
equal to the ladybug’s
cannot be determined 1 2 3 4 5

10. Tangential Acceleration
If the tangential speed changes, we can define a tangential acceleration

11. Rolling Motion
If a car rolls without slipping, then the distance the car travels = the arc length turned by the wheel
d = s = rθ
The condition for rolling w/o slipping can then be expressed as v = rω

12. How does the instantaneous velocity of the two points compare?
vA = vB
vA > vB
vA < vB
Cannot be determined B 1 2 3 4 5

13. Rolling w/o Slipping
The instantaneous velocity at the point of contact is 0
A perfect wheel has no kinetic friction
v = 2 rω
v = rω
v = 0

14. Ex. A car starts from rest and accelerates to 15 m/s in a time of 5
Ex. A car starts from rest and accelerates to 15 m/s in a time of 5.0 s. a) Sketch the situation. b) Find the car’s average acceleration during the 5.0 s. c) Find the angular velocity of the tires at t = 5.0 s if their diameter is 16”. d) Find the angular acceleration of the wheel during the 5.0 s.

15. Torque A torque produces an angular acceleration
Torque = τ = rF sinθ = Fperpendicular ∙r
If the torque produces a ccw rotation +,
cw - F r θ

16. Ex. A force of 10.0 N is applied along the tangent of a wheel of radius .50 m. a) sketch the situation. b) Find the net torque on the wheel.

17. Ex. Find the net force and torque for each situation shown below.
r = .25 m
r = .25 m
5.0 N
5.0 N

18. How do the magnitudes of the torques produced by identical forces for the two situations shown compare? A B
TA > TB
TA = TB
TA < TB
Cannot be determined 1 2 3 4 5

19. How do the magnitudes of the torques produced by identical forces for the two situations shown compare? B A
TA > TB
TA = TB
TA < TB
Cannot be determined 1 2 3 4 5

20. Torque Applied to a Rigid Body
A rigid body is a system where the particles are strongly bound to each other and maintain their relative orientation during motion
The net torque is the sum of all the indivisual torques applied to the system
τnet = Στi = ΣriFi sinθi

21. Newton’s Second Law Applied to Rotating Objects
τnet = Σri miai sin θi = Σri miat
at = rα
τnet = Σri mi ri α = Σmi ri2 α
τnet = Iα

22. Moment of Inertia Moment of Inertia is defined as I = Σmi ri2
Moment of Inertia resists changes in rotational motion
It depends not just on the mass but how far the mass is located from the axis of rotation

23. Ex. Find the moment of inertia for the following system of objects connected by light rods of the given length and rotating about the given axis.
.75 kg
.20 m
42°
1.0 kg
.40 m
.10 m
.25 kg
.075 m
.50 kg

24. A hoop and a disk (made of different materials) have identical masses and radii. Which has a greater moment of inertia?
Hoop
Disk
They are the same
Cannot be determined 1 2 3 4 5

25. Moments of Inertia of Extended Objects
Different shaped objects have different moments of inertia
In general, the further the mass is from the axis, the greater the moment of inertia

26. Identical torques are applied to a hoop and a disk of the same mass and radius. How do the resulting angular accelerations compare?
αhoop > αdisk
αhoop = αdisk
αhoop < αdisk
Cannot be determined 1 2 3 4 5

27. Four engines oriented 90° apart fire cw along the tangent of a 1200 kg cylindrical satellite of radius 2.5 m initially at rest. The engines each exert a thrust of 250 N. a) Sketch the situation. b) Draw a free body diagram for the satellite. c) What is the net force on the satellite? d) What is the net torque on the satellite? e) Find the angular acceleration of the satellite. f) If the rockets fire for 10.0 s, what is the angular velocity of the satellite?

28. Center of Gravity
The weight of an extended object can exert a torque on the object called the gravitational torque
τgrav = Σri mig = Mtotrcg g
We can calculate the torque as though it is due to all the weight Mtotg acting at a single point rcg called the center of gravity

29. Center of Gravity
The center of gravity of an object is the balance point. If the object is supported at the center of gravity, the gravitational torque on it is zero.
For a uniform object, the center of gravity coincides with the center of the object

30. Static Equilibrium
An object in static equilibrium has a velocity of 0 and an angular velocity of 0

31. Which of the following is true about an object in static equilibrium
Which of the following is true about an object in static equilibrium? (TPS)
Fnet = 0
τnet = 0
I = 0
1 and 2
1, 2, and 3 1 2 3 4 5

32. Which of the following is true about an object in static equilibrium
Which of the following is true about an object in static equilibrium? (TPS)
Fnet = 0
τnet = 0
I = 0
1 and 2
1, 2, and 3 1 2 3 4 5

33. Conditions for Static Equilibrium
Translational Equilibrium – Fnet = 0
Rotational Equilibrium – τnet = 0
If an object is in static equilibrium, then its net torque is 0 regardless about which axis the torque is calculated

34. Ex. A beam of negligible mass is supported at its center by a fulcrum
Ex. A beam of negligible mass is supported at its center by a fulcrum. A .50 kg mass hangs .30 m from one side of the support. And a .40 kg mass hangs from the other side of the beam. a) Sketch the situation. b) Draw a free body diagram for the beam. c) Find the distance form the center of the .40 kg beam so that equilibrium will result.

35. Ex. A uniform board of length 2. 0 m and mass 5
Ex. A uniform board of length 2.0 m and mass 5.0 kg is supported by a fulcrum at .35 m from one end. a) Sketch the situation. b) On which end must a mass be placed so that equilibrium results. c) How much mass must be placed at that end of the board so that equilibrium results?

36. Rotational Work and Energy
A spool with a string can be wound up to lift a weight
Rotating objects can do work

37. Rotational Kinetic Energy
In a rotating object each atom instantaneously moves along the tangent
Consequently, objects have kinetic energy when they rotate

38. Rotational Kinetic Energy
The total kinetic energy of a rotating object is KErot = Σ1/2 mivi2
But we can relate the tangential speed to the angular velocity vt = ω r
KErot = Σ1/2 miω2ri2 = (Σ1/2 miri2) ω2
KErot =1/2 Iω2

39. Rotational Work Energy Theorem
Rotational work changes rotational KE
τθ = Δ1/2 I ω2

40. FPE - Ex. A hoop of mass 0. 5 kg and radius
FPE - Ex. A hoop of mass 0.5 kg and radius .25 m is initially rotating at 20.0 s-1. Friction applies a torque of -2.0 Nm bringing the hoop to rest. a) Sketch the situation. b) Find the amount of work done by the friction. c) Through what angular displacement does the hoop turn?

41. Total Kinetic Energy of a Rolling Object
If an object slides it has KE = ½ mv2
If an object rotates it has KE = ½ Iω2
If an object is rolling it has both translational (sliding) and rotational kinetic energy
KErolling = ½ mv2 + ½ Iω2

42. A hoop and disk of the same mass and radius are released from rest at the top of a ramp. If both roll without slipping, which will reach the bottom of the ramp first? (TPS)
Hoop
Disk
Tie
Cannot be determined 1 2 3 4 5

43. A hoop and disk of the same mass and radius are released from rest at the top of a ramp. If both roll without slipping, which will reach the bottom of the ramp first? (TPS)
Hoop
Disk
Tie
Cannot be determined 1 2 3 4 5

44. A hoop and solid sphere of the same mass and radius are released from rest at the top of a ramp. If both roll without slipping, which will reach the bottom of the ramp first? (TPS)
Sphere
Hoop
Tie
Cannot be determined 1 2 3 4 5

45. A hoop and solid sphere of the same mass and radius are released from rest at the top of a ramp. If both roll without slipping, which will reach the bottom of the ramp first? (TPS)
Sphere
Hoop
Tie
Cannot be determined 1 2 3 4 5

46. A solid sphere and disk of the same mass and radius are released from rest at the top of a ramp. If both roll without slipping, which will reach the bottom of the ramp first? (TPS)
Sphere
Disk
Tie
Cannot be determined 1 2 3 4 5

47. A solid sphere and disk of the same mass and radius are released from rest at the top of a ramp. If both roll without slipping, which will reach the bottom of the ramp first? (TPS)
Sphere
Disk
Tie
Cannot be determined 1 2 3 4 5

48. Angular Momentum
The angular moment of a rigid body rotating about an axis is
L = Iω
SI Units?

49. Ex. A thin rectangular sheet of mass
Ex. A thin rectangular sheet of mass .50 kg and width perpendicular to the axis of .20 m rotates about an axis through its center at 15 s-1. a) Sketch the situation. b) Find the angular momentum of the board. c) What would we need to do to change the angular momentum of the board?

50. Angular Momentum of a System of Objects
If we have a system of objects than
Ltotal = ΣLi
To change the angular momentum of the system we exert a torque ΔL/Δt = τ
τtotal = Σ τi = Σ τi, ext + Σ τi, int
But Σ τi, int = 0 so τtotal = Σ τi, ext = τnet, ext

51. Conservation of Angular Momentum
ΔLtotal/Δt = τnet, ext
If τnet, ext = 0 then ΔLtotal/Δt = 0
If the net external torque on a system is 0, then the total angular momentum of the system is conserved

52. Ex. A figure skater is spinning on ice. a) Describe the system
Ex. A figure skater is spinning on ice. a) Describe the system. b) Are there any external torques acting on the system? c) If she brings her arms in, is she exerting an internal or external torque. d) What happens to her angular velocity when she brings her arms in? e) Explain in terms of conservation of angular momentum.

53. Ex. A disk of mass. 45 kg and radius. 20 m is initially spinning at
Ex. A disk of mass .45 kg and radius .20 m is initially spinning at .40 s-1 about an axis through the center of the circular face. A hoop of the same mass and radius is dropped from rest concentrically onto the disk so that the two stick together. a) Sketch the situation. b) Is the torque between the hoop and disk an external or internal torque to the system of hoop and disk? c) Is angular momentum conserved? d) Find the final angular velocity of the system.