1. Center of Mass Torque

2. Center of Mass When analyzing the motion of an extended object, we treat the entire object as if its mass were contained in a single point, known as the object’s center of mass (CM). Mathematically, the CM of an object is the weighted average of the location of the mass in an object.

3. 7-8 Center of Mass In (a), the diver’s motion is pure translation; in (b) it is translation plus rotation. There is one point that moves in the same path a particle would take if subjected to the same force as the diver. This point is the center of mass (CM).

4. 7-8 Center of Mass The general motion of an object can be considered as the sum of the translational motion of the CM, plus rotational, vibrational, or other forms of motion about the CM.

5. 7-8 Center of Mass The center of gravity is the point where the gravitational force can be considered to act. It is the same as the center of mass as long as the gravitational force does not vary among different parts of the object.

6. 7-8 Center of Mass The center of gravity can be found experimentally by suspending an object from different points. The CM need not be within the actual object – a doughnut’s CM is in the center of the hole.

7. Locating Center of Mass Center of mass can be “outside” an object.

8. 7-9 CM for the Human Body The x’s in the small diagram mark the CM of the listed body segments.

9. 7-9 CM for the Human Body The location of the center of mass of the leg (circled) will depend on the position of the leg.

10. 7-9 CM for the Human Body High jumpers have developed a technique where their CM actually passes under the bar as they go over it. This allows them to clear higher bars.

11. 7-10 Center of Mass and Translational Motion The total momentum of a system of particles is equal to the product of the total mass and the velocity of the center of mass. The sum of all the forces acting on a system is equal to the total mass of the system multiplied by the acceleration of the center of mass: (7-11)

12. 7-10 Center of Mass and Translational Motion This is particularly useful in the analysis of separations and explosions; the center of mass (which may not correspond to the position of any particle) continues to move according to the net force.

13. CM of Two Particles For two masses on a frictionless bar, where is the center of mass?

14. 7-8 Center of Mass For two particles, the center of mass lies closer to the one with the most mass: where M is the total mass.

15. General Formulas for CM M is total mass of system

16. Example m 1 =5 kg is located at x=10m from origin m 2 =10 kg is located at x=16m from origin

17. Do Now Find the center of mass of a system two masses if a 2 kg mass is located at x=0 and a 3 kg mass is located at x=10m. 2kg 3kg

18. Torque To make an object start rotating, a force is needed. If you push at the edge of the door with a force perpendicular to the door, the door rotates around the axes that passes through the hinge. The ability of a force to rotate an object around some axis is measured by a quantity called a torque.

19. 8-4 Torque The position and direction of the force that creates rotation is very important. The perpendicular distance from the axis of rotation to the line along which the force acts is called the lever arm.

20. Lever Arm To get the maximum torque, the force should be applied in a direction that creates the greatest lever arm.

21. 8-4 Torque Here, the lever arm for F A is the distance from the knob to the hinge; the lever arm for F D is zero; and the lever arm for F C is as shown.

22. Rate the forces using scale 1-3 according to their effectiveness in turning the nut? (1-most effective, 3- least effective).

23. 8-4 Torque A longer lever arm is very helpful in rotating objects.

27. Torque τ = Frsinθ r – distance from the axis of rotation to the point of the application of the force The symbol for torque is Greek letter tau SI unit of torque is Nm Newton-meter

28. Bar With Pivot The bar is balanced if the net torque =0 Pivot Weight of bar =20N Acts as if it was at the center of mass. What force should be applied at x to balance the torque? The length of the bar = 2m.

30. The Sign of a Torque Torque is a vector. If a positive torque is applied, the object will start rotating in a counter-clockwise direction. Negative torque produces clockwise rotation. Negative torque Positive torque

31. Balanced Torques Balance is achieved if the torque that tends to produce clockwise rotation by the boy equals the torque that tends to produce counterclockwise rotation by the girl.

32. Balanced Torques 10 N 20 N 1 m 0.5 m

33. Balanced Torques If the total torque on a motionless object is zero, the object will be balanced and not start rotating. Thus the sum of all torques on an object at equilibrium must be zero.

34. Balanced Torques 10 N 20 N 1 m 0.5 m Left torque = 10 N x 1 m = 10 N m Right torque = 20 N x 0.5 m = - 10 N m

35. Rotational Equilibrium When an object is in rotational equilibrium, the total torque applied to it is zero. Rotational equilibrium is often used to determine unknown forces.

36. What mass must be added to balance the scale? Find the force that balances the torque ?kg 2kg 5m 4m 1kg 2m

37. Example Find the force that balances the torque ? N 10N 5m 2m

38. Example A boy and his cat sit on a seesaw. The cat has a mass of 4 kg and sits 2 m from the center of rotation. If the boy has a mass of 50 kg, where should he sit so that the see-saw will balance?

39. Questions: 1) What is torque? 2)How do we calculate torque? 3) What are the units of measurement of torque? 4) What is rotational equilibrium?

40. Practice Problem What is the torque on a bolt applied with a wrench that has a lever arm of 45 cm with a force of 10 N? Torque = F x r = (10 N)(0.45m) = 4.5 N m