1. Dynamics of Rotational Motion
Chapter 10
Dynamics of Rotational Motion

2. Goals for Chapter 10
To learn what is meant by torque
To see how torque affects rotational motion
To analyze the motion of a body that rotates as it moves through space
To use work and power to solve problems for rotating bodies
To study angular momentum and how it changes with time

3. Loosen a bolt
Which of the three equal-magnitude forces in the figure is most likely to loosen the bolt?
Fa?
Fb?
Fc?

4. Loosen a bolt
Which of the three equal-magnitude forces in the figure is most likely to loosen the bolt?
Fb!

5. Let O be the point around which a solid body will be rotated.
Torque
Let O be the point around which a solid body will be rotated. O

6. Let O be the point around which a solid body will be rotated.
Torque
Line of action
Let O be the point around which a solid body will be rotated.
Apply an external force F on the body at some point.
The line of action of a force is the line along which the force vector lies. O
F applied

7. Let O be the point around which a body will be rotated.
Torque
Line of action
Let O be the point around which a body will be rotated.
The lever arm (or moment arm) for a force is The perpendicular distance from O to the line of action of the force L O
F applied

8. Torque
Line of action
The torque of a force with respect to O is the product of force and its lever arm t = F L
Greek Letter “Tau” = “torque”
Larger torque if
Larger Force applied
Greater Distance from O L O
F applied

9. Torque t = F L Larger torque if Larger Force applied
Greater Distance from O

10. Torque
Line of action
The torque of a force with respect to O is the product of force and its lever arm t = F L
Units: Newton-Meters
But wait, isn’t Nm = Joule?? L O
F applied

11. Torque Torque Units: Newton-Meters Use N-m or m-N
OR...Ft-Lbs or lb - feet
Not…
Lb/ft
Ft/lb
Newtons/meter
Meters/Newton

12. Torque
Torque is a ROTATIONAL VECTOR!
Directions:
“Counterclockwise” (+)
“Clockwise” (-)
“z-axis”

13. Torque
Torque is a ROTATIONAL VECTOR!
Directions:
“Counterclockwise”
“Clockwise”

14. Torque
Torque is a ROTATIONAL VECTOR!
Directions:
“Counterclockwise” (+)
“Clockwise” (-)
“z-axis”

15. Torque
Torque is a ROTATIONAL VECTOR!
But if r = 0, no torque!

16. Torque can be expressed as a vector using the vector cross product:
Torque as a vector
Torque can be expressed as a vector using the vector cross product:
t= r x F
Right Hand Rule for direction of torque.

17. Torque can be expressed as a vector using the vector cross product:
Torque as a vector
Torque can be expressed as a vector using the vector cross product:
t= r x F
Where r = vector from axis of rotation to point of application of Force
q = angle between r and F
Magnitude: t = r F sinq
Line of action
Lever arm O r q
F applied

18. Torque can be evaluated two ways:
Torque as a vector
Line of action
Torque can be evaluated two ways:
t= r x F t= r * (Fsin q)
remember sin q = sin (180-q)
Lever arm O r q
F applied
Fsinq

19. Torque can be evaluated two ways:
Torque as a vector
Line of action
Torque can be evaluated two ways:
t= r x F t= (rsin q) * F The lever arm L = r sin q
Lever arm
r sin q O q r
F applied

20. Torque as a vector Torque is zero three ways:
Line of action
Torque is zero three ways:
t= r x F t= (rsin q) * F t= r * (Fsin q)
If q is zero (F acts along r)
If r is zero (f acts at axis)
If net F is zero (other forces!)
Lever arm
r sin q O q r
F applied

21. Applying a torque
Find t if F = 900 N and q = 19 degrees

22. Applying a torque
Find t if F = 900 N and q = 19 degrees

23. Torque and angular acceleration for a rigid body
The rotational analog of Newton’s second law for a rigid body is z =Iz.
What is a for this example?

24. Torque and angular acceleration for a rigid body
The rotational analog of Newton’s second law for a rigid body is z =Iz.
t = Ia and t = Fr so Ia = Fr
a = Fr/I and a = r a

25. Another unwinding cable – Example 10.3
What is acceleration of block and tension in cable as it falls?

26. Rigid body rotation about a moving axis
Motion of rigid body is a combination of translational motion of center of mass and rotation about center of mass
Kinetic energy of a rotating & translating rigid body is KE = 1/2 Mvcm2 + 1/2 Icm2.

27. Rolling without slipping
The condition for rolling without slipping is vcm = R.

28. A yo-yo
What is the speed of the center of the yo-yo after falling a distance h?
Is this less or greater than an object of mass M that doesn’t rotate as it falls?

29. The race of the rolling bodies
Which one wins??
WHY???

30. Acceleration of a yo-yo
We have translation and rotation, so we use Newton’s second law for the acceleration of the center of mass and the rotational analog of Newton’s second law for the angular acceleration about the center of mass.
What is a and T for the yo-yo?

31. Acceleration of a rolling sphere
Use Newton’s second law for the motion of the center of mass and the rotation about the center of mass.
What are acceleration & magnitude of friction on ball?

32. Work and power in rotational motion
Tangential Force over angle does work
Total work done on a body by external torque is equal to the change in rotational kinetic energy of the body
Power due to a torque is P = zz

33. Work and power in rotational motion
Example 10.8: Calculating Power from Torque
Electric Motor provide 10-Nm torque on grindstone, with I = 2.0 kg-m2 about its shaft.
Starting from rest, find work W down by motor in 8 seconds and KE at that time.
What is the average power?

34. Work and power in rotational motion
Analogy to 10.8: Calculating Power from Force
Electric Motor provide 10-N force on block, with m = 2.0 kg.
Starting from rest, find work W down by motor in 8 seconds and KE at that time.
What is the average power?
W = F x d = 10N x distance travelled
Distance = v0t + ½ at2 = ½ at2
F = ma => a = F/m = 5 m/sec/sec => distance = 160 m
Work = 1600 J; Avg. Power = W/t = 200 W

35. Angular momentum Units = kg m2/sec (“radians” are implied!)
The angular momentum of a rigid body rotating about a symmetry axis is parallel to the angular velocity and is given by L = I.
Units = kg m2/sec (“radians” are implied!)
Direction = along RHR vector of angular velocity.  

36. Angular momentum For any system of particles  = dL/dt.
For a rigid body rotating about the z-axis z = Iz.
Ex 10.9: Turbine with I = 2.5 kgm2; w = (40 rad/s3)t2
What is L(t) & t = 3.0 seconds; what is t(t)?

37. Conservation of angular momentum
When the net external torque acting on a system is zero, the total angular momentum of the system is constant (conserved).
Follow Example using Figure below.

38. A rotational “collision”
Follow Example using Figure at the right.

39. Angular momentum in a crime bust
A bullet (mass = 400 m/s) hits a door (1.00 m wide mass = 15 kg) causing it to swing. What is w of the door?

40. Angular momentum in a crime bust
A bullet (mass = 400 m/s) hits a door (1.00 m wide mass = 15 kg) causing it to swing. What is w of the door?
Lintial = mbulletvbulletlbullet
Lfinal = Iw = (I door + I bullet) w
Idoor = Md2/3
I bullet = ml2
Linitial = Lfinal so mvl = Iw
Solve for w and KE final

41. Gyroscopes and precession
For a gyroscope, the axis of rotation changes direction. The motion of this axis is called precession.

42. Gyroscopes and precession

43. Gyroscopes and precession
For a gyroscope, the axis of rotation changes direction. The motion of this axis is called precession.

44. A rotating flywheel
Magnitude of angular momentum constant, but its direction changes continuously.

45. A precessing gyroscopic
Example 10.13