1. Torque What is the best way to loosen a bolt using a wrench?
Notice that where you exert the force matters!
Which of the following forces is most likely to loosen the bolt (assuming they all have the same magnitude):
Bolt
Line of action of force A
Point about which bolt rotates
Answer: Force C, because the tendency of a force to cause a rotation about some point depends on its magnitude and the perpendicular distance l between the line of action of the force and that point lB lC B
Line of action of force B C
Line of action of force C A
(lA = 0)

2. Torque
The distances lA, lB, and lC are called the lever arms (or moment arms) of forces A, B, and C (respectively) about the point of rotation
Torque (or moment) is a “twist” or a “turn” associated with a force and has a magnitude defined by: t = FlF
It is always defined with respect to a specific reference point
Note the similarity to force (a “push” or a “pull”)
Torque is the rotational analog of force
The torques associated with forces A, B, and C (see previous slide) with respect to the rotation point of the bolt are: tA = 0 , tB = BlB , and tC = ClC
Since the magnitudes of forces B and C are the same and lC > lB , then tC > tB

3. Torque
Forces B and C both tend to cause a counterclockwise rotation, which we can define to be the positive sense of rotation:
Thus tB and tC are both positive
However, a force directed in the opposite direction as B or C (tightening the bolt) would tend to cause a clockwise rotation, and the torque associated with this force would be negative
When a force acts in some arbitrary direction: + r f
Line of action of force F lF
lF = lever arm of force F = rsinf F f

4. Torque
The magnitude of the torque associated with force F is: t = FlF = Frsinf
Torque is a vector quantity  it has a magnitude and a direction
Direction is perpendicular to both the position ( ) and force ( ) vectors
The direction of torque is given by the Right-Hand Rule:
Point fingers of right hand in direction of position vector (begins at reference point and ends where acts)
Curl fingers toward direction of the force
Thumb points in direction of the torque
In the last example (see previous slide), torque points into of the page (or screen) – tends to cause clockwise rotation

5. CQ1: If all the forces below have equal magnitude, which one creates the most torque?
Figure A
Figure B
Figure C
Figure D

6. CQ2: A carpenter who is having a difficult time loosening a screw puts away his screwdriver and chooses another with a handle with a larger diameter. He does this because:
increasing force increases torque.
decreasing force decreases torque.
increasing lever arm increases torque.
decreasing lever arm decreases torque.

7. Center of Mass
Note that weight doesn’t act at a single point – it is distributed over the entire body
However, we can always calculate the torque due to a body’s weight by assuming that the entire force of gravity (weight) is concentrated at a single point called the center of gravity
If the acceleration of gravity has the same value at all points on a body, its center of gravity is identical to its center of mass (where all mass of object is considered concentrated at a single point)
A good assumption, considering that g at the bottom of the Sears Tower in Chicago is only about 0.014% greater than at the top
Center of gravity only about 2 cm below center of mass

8. Center of Mass For a system of point particles:
The center of mass of the system is the point in space having coordinates (xCM, yCM) defined by:
Average position of set of particles, weighted by their masses
Coordinates where all the mass of entire system can be considered to be concentrated m1 m2 m3

9. Center of Mass
Sometimes can use symmetry to deduce position of center of mass
Where would center of mass be for the previous system of 3 equally-spaced point masses if m1 = m2 = m3 ?
For solid bodies (which are a continuous distribution of matter), the sums given earlier would be impractical
Center of mass can usually be deduced from symmetry m1 m2 m3
center of mass
Homogeneous solid sphere:
Donut:
CM at center of sphere
CM at center of donut hole

10. CQ3: The system below consists of three spheres of equal mass m
CQ3: The system below consists of three spheres of equal mass m. The center of mass of the system is located at point: 3 4 5 6

11. Equilibrium of a Rigid Body
We can apply Newton’s 1st law two different ways to a rigid body when it is in equilibrium
For translational equilibrium:
Linear acceleration
For rotational equilibrium:
Angular acceleration

12. Example Problem #8.17
Consider the model at right of a person bending forward to lift a 200-N object. The spine and upper body are represented as a uniform horizontal rod of weight 350 N, pivoted at the base of the spine. The erector spinalis muscle, attached at a point 2/3 of the way up the spine, maintains the position of the back. The angle between the spine and this muscle is 12.0°. Find the tension in the back muscle and the compressional force in the spine.
2l/3 l
l/2 y + x
Solution (details given in class):
T = 2.71  103 N = 2.71 kN
Rx = 2.65 kN
(Note that when object is in equilibrium, you have freedom to choose the point about which to calculate the torques.)

13. CQ4: A sign hangs by a rope attached at 30° to the middle of its upper edge. It rests against a frictionless wall. If the weight of the sign were doubled, what would happen to the tension in the string? (Note: sin30° = 0.5; cos30° = 0.87)
It would remain the same.
It would increase by a factor of 1.5.
It would increase by a factor of 2.
It would increase by a factor of 4.

14. CQ5: A telephone pole stands as shown below
CQ5: A telephone pole stands as shown below. Line A is 4 m off the ground and line B is 3 m off the ground. The tensions in line A and line B are 200 N and 400 N respectively. What is the net torque on the pole?
0 Nm
400 Nm
800 Nm
2000 Nm

15. Example Problem #8.23
An 8.00-m, 200-N uniform ladder rests against a smooth wall. The coefficient of static friction between the ladder and the ground is 0.600, and the ladder makes a 50.0° angle with the ground. How far up the ladder can an 800-N person climb before the ladder begins to slip?
Solution (details in class):
6.15 m

16. Energy in Rotational Motion
Any matter in motion has kinetic energy
What is the kinetic energy of a rotating rigid body?
Representing the rigid body as the sum of individual point particles, the total kinetic energy of the body is the sum of the individual kinetic energies:
Each point particle has the same value of w, so:
Therefore, the kinetic energy of the rigid body is:

17. Moment of Inertia
Quantity = moment of inertia I of the body about a specific rotation axis (ri is distance from object i to the rotation axis)
Rotational kinetic energy:
Note similarity to translational (linear) kinetic energy (KT = ½ mv2)
Moment of inertia is the rotational analog of mass, and has to do with how mass is distributed
Consider 2 dumbbells:
Small moment of inertia  easy to start or stop rotation
Large moment of inertia  difficult to start or stop rotation

18. Moment of Inertia
For continuous distributions of matter (spheres, rods, cylinders, etc.), one would need methods of calculus to calculate moments of inertia
Moments of inertia of some common objects given in Table 8.1, p. 241 in textbook
Note that moment of inertia depends on how the mass is distributed and on the location of the axis of rotation

19. Total Mechanical Energy of Rigid Bodies
Rigid bodies can have rotational and/or translational (linear) kinetic energy
(vcm = velocity of center of mass)
The rigid body could also have gravitational potential energy
Ug = Mgycm (ycm = vertical position of center of mass)
In general, total mechanical energy of rigid body is thus: K1T + K1R + U1 = K2T + K2R + U2
If work is done by non-conservative forces (like friction): K1T + K1R + U1 + Wnc = K2T + K2R + U2

20. Newton’s 2nd Law for Rotation: Dynamics
Newton’s 2nd Law for linear motion:
Rotational analog for rigid bodies:
Notice the strong similarity to the linear version of Newton’s 2nd Law!
Newton’s 2nd Law for translation of a rigid body:
Depends on acceleration of the center of mass
Remember also the strong similarity between translational (KT) and rotational (KR) kinetic energy:
(For a rolling rigid body, I = Icm)

21. Example Problem: Ball Rolling Down an Incline
A ball of mass m and radius R starts from rest at a height of 2.00 m and rolls without slipping down a 30.0° slope as shown. Determine the linear acceleration of the ball down the incline. q
Solution (details given in class):
3.50 m/s2

22. Objects Rolling Down Incline Interactive
CQ6: In a lecture demonstration, Dr. Kaye “races” a solid sphere, a solid disk, and a thin cylindrical shell (hoop) by releasing them from rest at the top of an inclined plane of height h. In what order will the objects reach the bottom of the incline?
Sphere, disk, hoop
Sphere, hoop, disk
Disk, sphere, hoop
Disk, hoop, sphere
Hoop, sphere, disk
Hoop, disk, sphere
Objects Rolling Down Incline Interactive

23. Example Problem #8.40
An Atwood’s machine consists of blocks of masses m1 = 10.0 kg and m2 = 20.0 kg attached by a cord running over a pulley as shown. The pulley is a solid cylinder with mass M = 8.00 kg and radius r = m. The block of mass m2 is allowed to drop, and the cord turns the pulley without slipping. (a) Why must the tension T2 be greater than the tension T1? (b) What is the acceleration of the system, assuming the pulley axis is frictionless? (c) Find the tensions T1 and T2. T1 T2
Solution (details given in class):
2.88 m/s2
T1 = 127 N, T2 = 138 N

24. CQ7: Interactive Example Problem: The Flywheel Elevator
What is the student’s linear speed after falling 5.0 m, just before she reaches the ground?
1.0 m/s
5.0 m/s
10 m/s
25 m/s
(ActivPhysics Online Problem #7.10/7.12, Pearson/Addison Wesley)

25. Angular Momentum
The rotational analog of linear momentum is angular momentum, defined by:
Note similarity to linear relation p = mv
Angular momentum is a vector quantity
Direction is the same as w
Remembering Newton’s 2nd law written as: we have the following rotational analog:
If the net torque acting on a body is zero, then:
Angular momentum is constant in time, or conserved
Statement of conservation of angular momentum
Note similarity to conservation of linear momentum

26. Example Problem #8.60
A playground merry-go-round of radius 2.00 m has a moment of inertia I = 275 kgm2 and is rotating about a frictionless vertical axle. As a child of mass 25.0 kg stands at a distance of 1.00 m from the axle, the system (merry-go-round and child) rotates at the rate of 14.0 rev/min. The child then proceeds to walk toward the edge of the merry-go-round. What is the angular speed of the system when the child reaches the edge?
Solution (details given in class):
1.17 rad/s