Torque

2

Units: m N

Chap 8

Lever Arm is the vector from the point of application of a force to the axis of rotation (not necessarily the radius!).

F

F

F

Convention: CCW is positive (thumb is up)CW is negative (thumb is

down)

Which of the forces pictured as acting upon the rod will produce a torque about an axis perpendicular to the plane of the diagram

at the left end of the rod?

a) F1

b) F2 c) Both.d) Neither.

F2 will produce a torque about an axis at the left end of the rod. F1 has no lever arm with respect to the given axis.

The two forces in the diagram have the same magnitude. Which orientation will

produce the greater torque on the wheel?

F1 provides the larger torque. F2 has a smaller component perpendicular to the radius.

a) F1

b) F2 c) Both.d) Neither.

A 50-N force is applied at the end of a wrench handle that is 24 cm long. The force is applied in a direction perpendicular to the handle as shown. What is the torque applied to the nut by the wrench?

0.24 m 50 N = 12 N·m

a) 6 N·mb) 12 N·mc) 26 N·md) 120 N·m

What would the torque be if the force were applied half way up the handle instead of at the end?

0.12 m 50 N = 6 N·m

a) 6 N·mb) 12 N·mc) 26 N·md) 120 N·m

What is the torque about the axle of the merry-go-round due to the 50-N force?

-(1.2 m 50 N) = -60 N·m(clockwise)

a) +60 N·mb) -60 N·mc) +120 N·md) -120 N·m

What is the net torque acting on the merry-go-round?

96 N·m (counterclockwise) - 60 N·m (clockwise) = +36 N·m (counterclockwise)

a) +36 N·mb) -36 N·mc) +96 N·md) -60 N·me) +126 N·m

Equilibrium

We want to balance a 3-N weight against a 5-N weight on a beam. The 5-N weight is placed 20 cm to the right of a fulcrum. What is the torque produced by the 5-N weight?

a) +1 N·mb) -1 N·mc) +4 N·md) +4 N·m

F = 5 N = - Fll = 20 cm = 0.2 m = - (5 N)(0.2 m)

= -1 N·m

How far do we have to place the 3-N weight from the fulcrum to balance the system?

a) 2 cmb) 27 cmc) 33 cmd) 53 cm

F = 3 N l = / F = +1 N·m = (+1 N·m) / (3 N)

= 0.33 m = 33 cm

A constant net torque is applied to an object. Which one of the following will not be constant?

1. angular acceleration

2. angular velocity

3. moment of inertia

4. center of gravity

Quick Quiz

Center of Mass (CM)

You have learned have that:

rF perpendicular

But torque is also equal to:

IWhere and is measured in rad/s2tr

at

Is called MOMENT OF INERTIA (I), and is measured in kgm2

2rmI

r or l is the distance to the center of the rotation!!!!!!

2rmI

MOMENT OF INERTIA (I), measured in kgm2

Objects with different shapes, will have different I:

Objects with the same shape, but spinning around different axis, will have different I:

12

2MLI 12

2MLI

3

2MLI

2MRI

R

L

The only cases when you can actually calculate the MOMENT OF INERTIA (I) will be:

2mrI

When the object is very small, punctual (not an extended object), located at a distance r from the center of the rotation.Ex. a small ball, a piece of putty, etc.

Case A:

The only cases where you can actually calculate the MOMENT OF INERTIA (I) will be:

222 2mlmlmlI

Only if you can ignore the mass of the connecting rod:

Case B:

Only if you can ignore the mass of the connecting rod:

24

23

22

21 rmrmrmrmI

for all masses!

Case C:

Only if you can ignore the mass of the connecting rod:

244

233

222

211 rmrmrmrmI

031 rr

mrr 50.042

1

3

2 4

Case D:

Otherwise, consult the table in your book.

Newton’s 2nd Law for Rotation

• Draw free body diagrams of each object• Only the cylinder is rotating, so apply St = I a• The bucket is falling, but not rotating, so apply SF = ma• Remember that a = a r

The two rigid objects shown in the figure below have the same mass, radius, and angular speed. If the same braking torque is applied to each, which takes longer to stop?

Quick Quiz

1. A

2. B

3. more information is needed

Rotational Kinetic Energy (KEr)

2

2I

KEr

From now on, when bodies are TRANSLATING (moving along a straight line) AND ALSO ROTATING (such as a ball rolling down an incline), the Total Mechanic Energy (TME) will be:

PEKEKETME tr

mghmvI

TME 22

22

Unit: joule (J)

mghmvI

TME 22

22

Total KE

PEKEKEWork rt

Work done by external force F

( ) ( )t r g i t r g fKE KE PE KE KE PE If Work by external force is zero, then:

Two spheres, one hollow and one solid, are rotating with the same angular speed around an axis through their centers. Both spheres have the same mass and radius. Which sphere, if either, has the higher rotational kinetic energy?

1. the hollow sphere

2. the solid sphere

3. They have the same kinetic energy.

Quick Quiz

Angular Momentum (L)

ILL is measured in kgm2/s or also Js

t

L

t

I

tII

And, just like we’ve seen for linear momentum (p), the relation between L and torque is:

Angular momentum (L) is a vector.L has the direction of the change in torque.L is in the same axis as torque (parallel or antiparallel).L is in the same direction as angular velocity.

L is positive is counterclockwise.

L is negative if clockwise.

Conservation of Angular Momentum (L)

0net

finalinitial LL

ffii II

If

Then L is conserved:

Conservation of Angular Momentum

• Linear momentum is conserved if the net external force acting on the system is zero.

• Angular momentum (L) is conserved if the net external torque (τ) acting on the system is zero.

Conservation of Angular Momentum

How do spinning

skaters or divers

change their

rotational velocities

?

Change the

rotational inertia (I)

in order to

change the

rotational velocity (ω)

Conservation of Angular Momentum

Larger I (because R is larger) results insmaller ω, in order to keep L = I ω constant.

Smaller I (because R is smaller) results inlarger ω, in order to keep L = I ω constant.

Conservation of Angular Momentum

Change the

rotational inertia (I)

in order to

change the

rotational velocity (ω)

A horizontal disk with moment of inertia I1 rotates with angular speed ω1 about a vertical frictionless axle. A second horizontal disk, with moment of inertia I2 drops onto the first, initially not rotating but sharing the same axis as the first disk. Because their surfaces are rough, the two disks eventually reach the same angular speed ω. The ratio ω/ω1 is equal to

1. I1 / I2

2. I2 / I1

3. I1 / (I1 + I2)

4. I2 / (I1 + I2)

Quick Quiz