1. Torque and Center of Mass
Handout
HW #6

2. Center of Mass:
The center of mass (or mass center) is the mean location of all the mass in a system.
The motion of an object can be characterized by this point in space. All the mass of the object can be thought of being concentrated at this location. The motion of this point matches the motion of a point particle.

5. Finding the Center of Mass:
Uniform geometric figures have the center of mass located at the geometric center of the object.
Note that the center of mass does not have to be contained inside the volume of the object.

6. Collections of Point Masses:
The center of mass for a collection of point masses is the weighted average of the position of the objects in space.
Each object will have a position in space. The center of mass is found as:

7. Example #1: A 10. 0 kg mass sits at the origin, and a 30
Example #1: A 10.0 kg mass sits at the origin, and a 30.0 kg mass rests at the 12.0 m mark on the x – axis. (a) Find the center of mass for this system.

8. (b) Find the center of mass for this system relative to the mass at the right.
Although numerically different, it is the same point in space relative to the masses…

9. Example #2: A 10. 0 cm long wire has a mass of 4. 00 grams
Example #2: A 10.0 cm long wire has a mass of 4.00 grams. This wire is bent into an “L” shape that measures 6.00 cm by 4.00 cm, as shown below. Determine the center of mass for this object.

10. Treat as two objects: 6 cm object: 4 cm object:
Example #2: A 10.0 cm long wire has a mass of 4.00 grams. This wire is bent into an “L” shape that measures 6.00 cm by 4.00 cm, as shown below. Determine the center of mass for this object.
Treat as two objects:
6 cm object:
4 cm object:

12. Example #3: Determine the center of mass of the following masses, as measured from the left end. Assume the blocks are of the same density.

15. Torque
Torque is the rotational equivalent of force. A torque is the result of a force applied to an object that tries to make the object rotate about some pivot point.

16. Note that torque is maximum when the angle q is 90º.
Equation of Torque:
applied force
distance from pivot to applied force
angle between direction of force and pivot distance.
pivot point
Note that torque is maximum when the angle q is 90º.
The units of torque are Nm or newton · meter

17. The torque is also the product of the distance from the pivot times the component of the force perpendicular to the distance from the pivot.

18. The torque is also the product of the force times the lever arm distance, d.

19. Example #4: Calculate the torque for the force shown below.

20. Example #5: Calculate the total torque about point O on the figure below. Take counterclockwise torques to be positive, and clockwise torques to be negative.

21. Example #6: The forces applied to the cylinder below are F1 = 6
Example #6: The forces applied to the cylinder below are F1 = 6.0 N, F2 = 4.0 N, F3 = 2.0 N, and F4 = 5.0 N. Also, R1 = 5.0 cm and R2 = 12 cm. Determine the net torque on the cylinder.