Equilibrium Lecturer: Professor Stephen T. Thornton.

Equilibrium

Lecturer: Professor Stephen T. Thornton

Reading QuizReading QuizA) all

B) 1 only

C) 2 only

D) 3 only

E) 2 and 3

A box is placed on a ramp in the

configurations shown below. Friction

prevents it from sliding. The center of

mass of the box is indicated by a blue

dot in each case. In which case(s) does

the box tip over?

1 2 3

Reading QuizReading Quiz

1 2 3

A) all

B) 1 only

C) 2 only

D) 3 only

E) 2 and 3

A box is placed on a ramp in the

configurations shown below. Friction

prevents it from sliding. The center of

mass of the box is indicated by a blue

dot in each case. In which case(s) does

the box tip over?

The torque due to gravity acts

like all the mass of an object is

concentrated at the CM.

Consider the bottom right corner

of the box to be a pivot point.

If the box can rotate such that If the box can rotate such that

the CM is lowered, it will!!the CM is lowered, it will!!

Last Time

Angular momentum and torque

Gyroscopes

Inertial forces, pseudoforces

Review

Today

Force equilibrium

Rotational equilibrium

Stability and balance

Elasticity; stress and strain

Copyright © 2009 Pearson Education, Inc.

The Conditions for (force) Equilibrium

0ii

F

Copyright © 2009 Pearson Education, Inc.

The second condition of equilibrium is that there be no torque around any axis; the choice of axis is arbitrary.

The Conditions for (torque) Equilibrium

0ii

Forces Required for Equilibrium

1 2 force equilibriumF F mg

Is the plank in equilibrium?The forces may be balanced, but is the torque balanced? Maybe not.

Condition for force equilibrium: 0, 0 x yF F 0i

Look carefully at plank problem. Make sure you understand how to get the torques. Find torque about any pivot.

Condition for rotational equilibrium:

2

30

4

LF L mgt = - =

A Lamp in Equilibrium

; forces

torque

y xf mg f T

TV mgH

Do rotational equilibrium demo.

Zero Torque and Balance

This is primarily a torque problem. Why is that?

Zero Torque and Balance

1 1 2 2g gm x m x

This is primarily a torque problem. Why is that?

1 2m g m gT +=

The Geometric Center of the United States

Do CM of Virginia demo again.

Conceptual Quiz:A box, with its CM off center as indicated by the dot, is placed on an inclined plane. In which of the four orientations shown, if any, does the box tip over?

A) AB) BC) CD) DE) Box does not tip

over.

A

DC

B·

··

·

Answer: C

This is the only orientation where the CM is to the left of the likely pivot point on the bottom left corner of the box.

·

··

·

Conceptual QuizConceptual Quiz

1kg

1m

A 1-kg ball is hung at the end of a rod

1-m long. If the system balances at a

point on the rod 0.25 m from the end

holding the mass, what is the mass of

the rod?

A) kg

B) kg

C) 1 kg

D) 2 kg

E) 4 kg

12

14

1 kg

X

CM of rod

same distance mROD = 1 kg

A 1-kg ball is hung at the end of a rod

1-m long. If the system balances at a

point on the rod 0.25 m from the end

holding the mass, what is the mass of

the rod?

The total torque about the pivot The total torque about the pivot

must be zero !!must be zero !! The CM of the

rod is at its center, 0.25 m to 0.25 m to

the right of the pivotthe right of the pivot. Because

this must balance the ball,

which is the same distance to same distance to

the left of the pivotthe left of the pivot, the masses

must be the same !!

A) kg

B) kg

C) 1 kg

D) 2 kg

E) 4 kg

Conceptual QuizConceptual Quiz

12

14

Answer: C

0.25 m of stick on each side cancels out. Next 0.5 m of stick must balance rock. Distance is 2X, mass in second half of stick must be ½ kg. Total mass of stick is then 1 kg.

1 kg0.5 kg

1. Choose one object at a time, and make a free-body diagram by showing all the forces on it and where they act.

2. Choose a coordinate system and resolve forces into components.

3. Write equilibrium equations for the forces.

4. Choose any axis perpendicular to the plane of the forces and write the torque equilibrium equation. A clever choice here can simplify the problem enormously.

5. Solve.

Solving Statics Problems

If a force in your solution comes out negative (as A will here), it just means that it’s in the opposite direction from the one you chose. This is trivial to fix, so don’t worry about getting all the signs of the forces right before you start solving.

Solving Statics Problems

F

Propped Board. A large 62.0-kg board is propped at a 45° angle against the edge of a barn door that is 2.6 m wide. How great a horizontal force must a person behind the door exert (at the edge) in order to open it? Assume that there is negligible friction between the door and the board but that the board is firmly set against the ground.

If the forces on an object are such that they tend to return it to its equilibrium position, it is said to be in stable equilibrium.

Stability and Balance

If, however, the forces tend to move it away from its equilibrium point, it is said to be in unstable equilibrium.

Stability and Balance

Copyright © 2009 Pearson Education, Inc.

An object in stable equilibrium may become unstable if it is tipped so that its center of gravity is outside the pivot point. Of course, it will be stable again once it lands!

Stability and Balance

Copyright © 2009 Pearson Education, Inc.

People carrying heavy loads automatically adjust their posture so their center of mass is over their feet. This can lead to injury if the contortion is too great.

Stability and Balance

Demos: CM toys

Do demo of blocks on edge of table.

Conceptual QuizConceptual Quiz

1 kg

1 m 3 m

1 m 2 m

?

?

A (static) mobile hangs as shown

below. The rods are massless and

have lengths as indicated. The mass

of the ball at the bottom right is 1 kg.

What is the total mass of the mobile?

A) 5 kg

B) 6 kg

C) 7 kg

D) 8 kg

E) 9 kg

1 kg

1 m 3 m

1 m 2 m

?

?

A (static) mobile hangs as shown

below. The rods are massless and

have lengths as indicated. The mass

of the ball at the bottom right is 1 kg.

What is the total mass of the mobile?

Use torques in two steps: (1)

find the big mass on the bottom

left (lower rod only), and (2) use

the entire lower rod assembly

(with two masses) to find the

mass on top right. Finally, add Finally, add

up all the masses.up all the masses.

Conceptual QuizConceptual QuizA) 5 kg

B) 6 kg

C) 7 kg

D) 8 kg

E) 9 kg

Conceptual QuizConceptual Quiz

A) case 1 will tip

B) case 2 will tip

C) both will tip

D) neither will tip

Consider the two configurations of

books shown below. Which of the

following is true?

1/2

1/4 1/2

1/4

1 2

Conceptual QuizConceptual Quiz

The CM of the system is

midway between the CM of

each book. Therefore, the

CM of case #1 is not over the CM of case #1 is not over the

table, so it will tiptable, so it will tip.

A) case 1 will tip

B) case 2 will tip

C) both will tip

D) neither will tip

Consider the two configurations of

books shown below. Which of the

following is true?

1/2

1/4 1/2

1/4

1 2