1. Torque and Rotation
Physics

2. Torque Force is the action that creates changes in linear motion.
For rotational motion, the same force can cause very different results.
A torque is an action that causes objects to rotate.
A torque is required to rotate an object, just as a force is required to move an object in a line.

3. Torque is created by force, but it also depends on where the force is applied and the point about which the object rotates.
For example, a door pushed at its handle will easily turn and open, but a door pushed near its hinges will not move as easily. The force may be the same but the torque is quite different.

4. Torque
Torques change angular velocity. The symbol for torque is the Greek letter . Torque is given by this equation: τ = rF or
r is the distance to the center of spin from where the force is applied.
This variable is often called the lever arm.
F sin  be the force component that is perpendicular to the lever arm

5. Components of Torque

6. Torque Units The unit for torque is going to be a newton meter(Nm).
This looks very similar to the unit for work, the joule, but it is quite different.
So energy and work are in joules and torque is left in Newton meters.

7. Center of rotation
The point or line about which an object turns is its center of rotation.
For example, a door’s center of rotation is at its hinges.
A force applied far from the center of rotation produces a greater torque than a force applied close to the center of rotation.

8. Line of action
Torque is created when the line of action of a force does not pass through the center of rotation.

9. Force applied must be perpendicular
The lever arm is the perpendicular distance between the line of action of the force and the center of rotation

10. Calculating torque
The torque (τ) created by a force is equal to the lever arm (r) times the magnitude of the force (F).

11. Torques can be added and subtracted
If more than one torque acts on an object, the torques are combined to determine the net torque. If the torques tend to make an object spin in the same direction (clockwise or counterclockwise), they are added together.
If the torques tend to make the object spin in opposite directions, the torques are subtracted.

12. Adding Torques

13. Reminder: Units of torque
The units of torque are force times distance, or newton-meters.
A torque of 1 N-m is created by a force of 1 newton acting with a lever arm of 1 meter.

14. You try…
A force of 50 newtons is applied to a wrench that is 30 centimeters long.
Calculate the torque if the force is applied perpendicular to the wrench so the lever arm is 30 cm.
Set up the information:
Lets draw a picture with the given information.
Go back into the notes and find the equation for torque.
Analyze the information: Plug in the information. Are the units correct?
Test the information . Does it make sense????
( Remember downward forces are negative)

15. You try…
A force of 50 newtons is applied to a wrench that is 30 centimeters long.
Calculate the torque if the force is applied perpendicular to the wrench so the lever arm is 30 cm.
1) You are asked to find the torque.
2) You are given the force and lever arm.
3) The formula that applies is τ = rF.
4) Solve:
τ = (-50 N)(0.3 m) = -15 N.m

16. Net torque is zero
When an object is in rotational equilibrium, the net torque applied to it is zero.
For example, if an object such as a see-saw is not rotating, you know the torque on each side is balanced

17. Unknown forces
Rotational equilibrium is often used to determine unknown forces.
Any object that is not moving is in rotational equilibrium and in translational equilibrium.

18. Example
For example, consider a 10-meter bridge that weighs 500 newtons supported at both ends. A person who weighs 750 newtons is standing 2 meters from one end of the bridge.
What are the forces (FA, FB) holding the bridge up at either end?

19. Example
For example, consider a 10-meter bridge that weighs 500 newtons supported at both ends. A person who weighs 750 newtons is standing 2 meters from one end of the bridge.
What are the forces (FA, FB) holding the bridge up at either end?
For the bridge not to move up or down, the total upward force must equal the total
downward force. This means FA + FB = 1,250 N. Unfortunately, balanced force in
the vertical direction does not tell you how the force is divided between the two
ends, FA and FB.
Solving for the
unknown forces
For the bridge to be in rotational equilibrium, the total torque around any point
must be zero. If we choose the left end of the bridge, the torque created by force
FA is zero because its line of action passes through the center of rotation. By
setting the total of the remaining torques to zero, the force on the right support
(FB) is calculated to be 400 newtons. Since the total of both forces must be 1,250
N, that means the force on the left (FA) must be 850 N. This kind of analysis is
used to solve many problems in physics and engineering, including how strong to
make bridges, floors, ladders, and other structures that must support forces.

20. You Try…
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?

21. A boy and his cat sit on a seesaw
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?

22. Force and lever arm are not always perpendicular
When the force and lever arm are not perpendicular, an extra step is required to calculate the length of the lever arm.

23. You try…
A 20-centimeter wrench is used to loosen a bolt. The force is applied 0.20 m from the bolt. It takes 50 newtons to loosen the bolt when the force is applied perpendicular to the wrench. How much force would it take if the force was applied at a 30-degree angle from perpendicular?

24. Let’s look at another example
125 N is applied to a nut by a wrench. The length of the wrench is m. What is the torque?

25. A torque of 857 Nm is applied to flywheel that has a radius of 45.5 cm. What is the applied force?

26. MULTIPLE TORQUES
What happens if two or more torques act on an object at the same time?
Two forces are applied to the object. The object is free to rotate about the spin axis. Both cause a torque.

27. F1 causes a CCW (counter clockwise) rotation around the axis.
F2 causes a CW (clockwise) rotation around the axis.
If a torque causes a clockwise rotation, it is positive.
If a torque causes a counter clockwise rotation, it is negative.
The sum of the two torques would be:

28. Equilibrium and Torque:
If an object is in angular equilibrium (sometimes called rotational equilibrium), then it is either at rest or else it is rotating with a constant angular:
If object is in rotational equilibrium, the net torque about any axis is zero
  = 0

29. The net force must be zero and the net torque must be zero.
Static equilibrium exists when an object has no motion, either linear or angular. There are two conditions which must exist in order to have your good old static equilibrium:
The net force must be zero and the net torque must be zero.
 F = 0
  = 0

30. This gives us some very powerful tools to solve static problems
This gives us some very powerful tools to solve static problems. We can analyze a system and look at the forces acting on it, and we can also look at the torques that act on it. We’ll be able to do some really cool stuff.
For example:

31. Two metal orbs are attached to a very lightweight rigid wire
Two metal orbs are attached to a very lightweight rigid wire. They are suspended from a rigid point on the overhead as shown. The system does not move. Calculate the distance from the suspension line to the center of gravity on the right sphere.

33. Without using the torque equilibrium, we could not solve the problem
Without using the torque equilibrium, we could not solve the problem. The sum of forces would simply tell us that the upward force would be equal to weight of the two balls.
Using torque, however, allows us to solve the problem.
All we have to do is add up d’ torques:

34. SOLUTION

35. Watch this video clip http://www.youtube.com/watch?v=-iS4XH6hcqs

36. Which force causes the largest magnitude torque? A) A B) B C) C
Three forces labeled A, B, C are applied to a rod which pivots on an axis thru its center
[ ]
Which force causes the largest magnitude torque?
A) A B) B C) C
D) two or more forces tie for largest size torque.
Answer: A.
For C L/4 * 2F =L/2*F , . For B, torque = L/2 *-F .
For A, Lfsin 45 degrees=.707LF . 

37. The force exerted on the door by the hinge... A) is zero
A door is pushed on by two forces, a smaller force at the door knob and a larger force nearer the hinge as shown. The door does not move.
The force exerted on the door by the hinge...
A) is zero
B) points  (along +y)
C) points (along -y)
D) points (lower right, in diagram)
E) points in some other direction
Answer C: points lower right, in diagram.
Since the door is not moving, we must have . The small force has a negative x-component, so the hinge force must have a positive x-component to cancel it. The small force has a small negative y-component, while the big force has a big positive y-component. The hinge force must have a positive y-component to help the

38. (Hint: consider the torque about the mass M).
A mass M is placed on a very light board supported at the ends, as shown. The free-body diagram shows directions of the forces, but not their correct relative sizes.
What is the ratio ?
(Hint: consider the torque about the mass M).
A) 2/3 B) 1/3 C) 1/2 D) 2
E) some other color.
Answer: 2. The total torque about the point where M rests must be zero, otherwise the board would start rotating. Since the lever arms are in the ratio 1 to 2 [(1/3)L to (2/3)L], the force must be in the ratio 2 to 1 so the torques are equal magnitude.
Suppose now the board has mass m, so the free-body diagram is now:
Compared to when the board had no mass, the force FR is now ..
A: greater B: less
C: the same.
Answer: greater, FR increases as m increases.
If the board is in equilibrium, we must have . This tells us that (FL+FR­) = (m+M)g. So if m increases, the sum (FL+FR) increases. But we still don't know for sure that FR increases. To see that FR­ must increase, consider the torques about the left end of the board. We must have . So as mg increases, FR must increase so the total torque remains zero.

39. A planet in elliptical orbit about the Sun is in the position shown.
With the origin located at the Sun, the vector torque on the planet..
A) is zero. B) points along +z.
C) is in the x-y plane. D) None of these.
With the origin located at the Sun, the vector torque on the planet..
Pink: is zero.     Yellow: points along +z.    Green: is in the x-y plane.     Purple: None of these.
Answer: torque is zero! The vectors are anti-parallel. So q =180, sinq =0 in t =rRsinq . Central-forces always give zero torque. 

40. Which one experiences the larger torque?
Two light (massless) rods, labeled A and B, each are connected to the ceiling by a frictionless pivot. Rod A has length L and has a mass m at the end of the rod. Rod B has length L/2 and has a mass 2m at its end. Both rods are released from rest in a horizontal position.
Pink: A Yellow: B Purple: Both have the same size .
Answer: Both have the same size torque.
t A = RFsinq = Lmg sinq.
t B = RFsinq = (L/2)2mg sinq= Lmg sinq
Which one experiences the larger torque?
A) A B) B C) Both have the same size .

41. Which one falls to the vertical position fastest?
Two light (massless) rods, labeled A and B, each are connected to the ceiling by a frictionless pivot. Rod A has length L and has a mass m at the end of the rod. Rod B has length L/2 and has a mass 2m at its end. Both rods are released from rest in a horizontal position.
Answers: Both have the same torque. And B falls fastest.
The one with the smallest I has the largest  (since both have the same torque). IA = mL2, IB = 2m (L/2)2 = (1/2) mL2. IB has a smaller I, a larger , so it falls faster.
Which one falls to the vertical position fastest?
A) A B) B C) Both fall at the same rate
Hint