1. Chapter 11 Rotational Dynamics and Static Equilibrium

2. Chapter 11: Rotational Dynamics and Static Equilibrium
Torque: The ability of a force to rotate a body about some axis t = rF Note: F  r
The torque is larger if the force is applied farther from the axis of rotation.

3. By convention, the sign of torque is:
t is negative clockwise (cw)
t is positive counter-clockwise (ccw)

4. General Definition of Torque
Only the component of the force that is perpendicular to the radius causes a torque.
= r(Fsinq)
Equivalently, only the perpendicular distance between the line of force and the axis of rotation, known as the moment arm r, can be used to calculate the torque.
t = rF = (rsinq)F F q
Fsinq

5. Each force that acts on an object may cause a torque.
In this figure, the three forces have equal magnitude.
Which forces cause a torque?
Which force causes the biggest magnitude torque?
Which forces, if any, causes a positive torque? F3 r2 r1
pivot point F2
When discussing torques, we must identify a pivot point (or axis of rotation).
The net torque about a point O is the sum of all torques about O: St = t1 + t

6. HW 11 problem # 1
A person holds a 1.42 N baseball in his hand, a distance of 2L = 34 cm from the elbow joint, as shown in the figure. The biceps, attached at a distance of d = 2.75 cm from the elbow, exert an upward force of 12.8 N on the forearm. Consider the forearm and hand to be a uniform rod with a mass of 1.39 kg.
(a) Calculate the magnitude of the net torque acting on the forearm and hand. Use the elbow joint as the axis of rotation. [2.44 N.m]
(b) If the net torque obtained in part (a) is nonzero, in which direction will the forearm and hand rotate?
[clockwise]

7. Moment of Inertia
Recall that mass (inertia) is an object’s resistance to acceleration. Similarly an object’s resistance to rotation (angular acceleration) is known as moment of inertia. For a point mass m:
I = mr2
I = moment of inertia
r = distance from the axis of rotation
For an extended object:
I =Smiri2
Mass near the axis of rotation resists rotation less than mass far from the axis of rotation.

8. Hoop or Cylindrical Shell Solid Cylinder or Thin Disk
Solid Sphere                      
Spherical Shell                      
Hoop or Cylindrical Shell                      
Solid Cylinder or Thin Disk                        
Thin Rod or Bar                                  
Thin Rod about its end                                  

9. Angular Position, q
For circular motion, the distance (arc length) s, the radius r, and the angle  are related by:
q > 0 for counterclockwise rotation from reference line
Note that  is measured in radians:
1 rev = 360° = 2p rad

10. Consider a rotating disk:P r s r  O O P
t = 0
t > 0

11. which leads to the average angular speed wav
Angular Velocity, w
Notice that as the disk rotates,  changes. We define the angular displacement, , as:
 = f - i
which leads to the average angular speed wav

12. Period
The period of rotation is the time it takes to complete one revolution.
T = period
Rearranging we have
What is the period of the Earth’s rotation about its own axis?
What is the angular velocity of the Earth’s rotation about its own axis?

13. Angular Acceleration, a
We can also define the average angular acceleration aav:
and
The SI units of a are: rad/s2 = s-2
We will skip any detailed discussion of angular acceleration, except to note that angular acceleration is the time rate of change of angular velocity

14. Torque and Angular Acceleration
Recall Newton’s Second Law: F = ma
The net force on an object of mass m causes a (linear) acceleration a.
Similarly, the net torque on an object with moment of inertia I causes an angular acceleration a.
t = Ia

15. HW11 - Problem
When a ceiling fan rotating with an angular speed of 2.15 rad/s is turned off, a frictional torque of N m slows it to a stop in 6.25 s. What is the moment of inertia of the fan? [0.701] kg m2

16. Zero Torque and Static Equilibrium
Consider the wheel shown below. Two forces of equal magnitude are acting on the wheel. Will the wheel remain at rest?
The net force is zero, so there will be no linear acceleration. F1 F2
However, the sum of the torques is not zero, so there will be an angular acceleration.
The wheel is not in static equilibrium.

17. Conditions for Static Equilibrium
For true static equilibrium, two conditions must be satisfied:
For an object in equilibrium, the axis of rotation is arbitrary (But all torques must be evaluated about a common axis).

18. Angular Momentum For linear momentum: p = mv
For rotational motion, we define an angular momentum:
L = Iw
The SI units of angular momentum are kg·m2/s

19. Angular Momentum - Problem
A kg record with a radius of 15 cm rotates with an angular speed of 29 rpm. Find the angular momentum of the record. [4.44E-4] kg m2/s

20. Kinetic energy of rotation
What is the kinetic energy of a mass m traveling at speed v in a circle of radius r?
K = (1/2) m v2 = (1/2) mr2 (v/r) 2 = (1/2) I w2
Kinetic energy of rotation = (1/2) I w2
This is not a new form of energy, just a re-labeling (or alternate formula) for kinetic energy.

21. Rotational Kinetic Energy - Problem
Calculate the rotational kinetic energy of the Earth as it (a) orbits the sun (b) rotates about its axis.
Mass of Earth = 5.98E24 kg
Radius of Earth (ave) = 6.38E6 m
Average Earth-Sun distance = 1.50E11 m