What’s the lever arm between F and the fulcrum?

A. l

B. a

C. b

ab

2

When the applied force is not perpendicular to the crowbar, for example, the lever arm is found by drawing the perpendicular line from the fulcrum to the line of action of the force.

We call torques that produce counterclockwiserotation positive, and torques that produce clockwise rotation negative.

3

Two forces are applied to a merry-go-round with a radius of 1.2 m as shown. What is the torque about the axle of the merry-go-round due to the 80-N force?

1.2 m 80 N = +96 N·m

(counterclockwise)

a) +9.6 N·m

b) -36 N·m

c) +96 N·m

d) -36 N·m

4

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·m

b) -60 N·m

c) +120 N·m

d) -120 N·m

5

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·m

b) -36 N·m

c) +96 N·m

d) -60 N·m

e) +126 N·m

6

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

7

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

8

is the point about which the weight of the object itself exerts no torque.

We can locate the center of gravity by finding the point where it balances on a fulcrum.

What’s the center of gravity of a disk? A. any point on the edge of the disk. B. Center of the disk C. Any point half way between the center and the edge.

The center of gravity

9

For a more complex object, we locate the center of gravity by suspending the object from two different points, drawing a line straight down from the point of suspension in each case, and locating the point of intersection of the two lines.

The center of gravity

10

1110/9/2019 Physics 214

Fall 2009

11

Where is the Center of Gravity of the cut

board ?

1J-21 Center of Gravity of an Irregular Lamina

How can we find the Center of Gravity of Irregular shapes?

Why does the mass on the string hang

straight down ?

If the center of gravity lies below the pivot point, the object will automatically regain its balance when disturbed.

The center of gravity returns to the position directly below the pivot point, where the weight of the object produces no torque.

12

1310/9/2019 Physics 214

Fall 2009

13

How difficult is it to balance this system on

a sharp point ?

1J-23 Corks & Forks

THE CENTER OF GRAVITY IS NOT ON THE OBJECT. IT ACTUALLY LIES ALONG THE VERTICAL BELOW THE SHARP POINT.WHEN THE FORK IS MOVED THE CM RISES AND THIS MEANS THE SYSTEM IS IN STABLE EQUILIBRIUM .

Can the Center of Gravity lie at a point not on the object?

Where is the C of G ?

1410/9/2019 Physics 214

Fall 2009

14

Can you safely walk to the end of the

plank ?

1J-16 Walk the Plank

EVEN WITH A MASS AT THE END OF THE PLANK, THE SYSTEM CAN STILL BE IN EQUILIBRIUM

Sum of Torque about Pivot

X M g – x m g = 0

m = M X / x

What happens when a mass is placed at the end of a massive plank?

One can solve for either M or m, if

the other quantity is known

1510/9/2019 Physics 214

Fall 2009

15

1m1x 2x 2m

M

How does this system Balance?

1J-28 Wine Bottle Holder

THE CENTER OF GRAVITY OF THE BOTTLE PLUS THE WOOD MUST LIE DIRECTLY OVER AND WITHIN THE BOUNDARY OF THE SUPPORT (PIVOT). FOR BALANCE THERE CAN BE NO NET TORQUE ON SYSTEM.

Sum of Torque about Pivot

m1x1g - m2x2g = 0

Balance a Bottle and a Wooden Holder by Eliminating Net Torque

16

1J - 24 double cone

The force causing the object to move is gravity and we know that by energy conservation that if the object gains kinetic energy it must lose potential energy. Therefore the center of mass must be falling and the kinetic energy = mgh where h is the distance the CM falls.

What happens to the center of mass ?

A). Going down the hillB). Going uphill C). Stay at restD). Depend on the object shape

Quiz: where is the CG of the beam and the CG of the system (beam + two weights) after it’s balanced?

a) CG of the beam is Center of the beam, CG of the system is at the pivot point.

b) Both are at the center of the beam.

c) Both are the pivot points.

17

How far can the child walk without tipping

the plank?

• For a uniform plank, its center of gravity is at its geometric center.• The pivot point will be the edge of the supporting platform.• The plank will not tip as long as the counterclockwise torque from the

weight of the plank is larger than the clockwise torque from the weight of the child.

• The plank will verge on tipping when the magnitude of the torque of the child equals that of the plank.

18

An 80-N plank is placed on a dock as shown. The plank is uniform in density so the center of gravity of

the plank is located at the center of the plank. A 150-N boy standing on the plank walks out slowly from the edge of the dock. What is the torque

exerted by the weight of the plank about the pivot point at the edge of the dock?

a) +80 N·mb) -80 N·mc) +160 N·md) -160 N·me) +240 N·mf) -240 N·m

1 m 80 N = +80 N·m

(counterclockwise)

19

Quiz: An 80-N plank is placed on a dock as shown. The plank is uniform in density so the center of gravity of the plank is located at the center of the plank. A 150-N boy standing on the plank walks out slowly from the edge of the dock. How far from the edge of the dock can the 150-N boy walk until

the plank is just on the verge of tipping?

a) 0.12 m

b) 0.23 m

c) 0.53m

d) 1.20 m

80 N·m / 150 N = 0.53 m

20