4.3. Tension and Pulleys

Demonstration

1. Calibrate 2 spring scales, put them

together horizontally, and pull apart.

Do they always show the same force?

What are they measuring?

2. Put 2 spring scales together vertically,

and calibrate both to zero. If a weight

is added at the bottom, will both scales

show that weight, or half that weight?

Why?

Strings and Tension

When you pull on a string or rope, it becomes

taut. We say that there is tension in the string.

Tension is a force. How does its magnitude and

direction vary along the string?

Strings and Tension

If the string were cut in any place, equal and opposite

forces would be needed to hold the string together again.

All along an uncut string are equal and opposite forces, for

no net acceleration.

Demonstration

• Have two people hold a string under high

tension. Have a third person cut the string

in half. What happens? Why?

Strings and Tension

In a sense, the string transmits forces from one

end to the other. Tie a string to a chair, and pull.

The force of the chair’s resistance is transmitted

to your hand, and you hand’s force to the chair.

Strings and Tension

The tension in a real rope will vary along its

length due to the weight of the rope itself.

Here, we will assume that

all ropes, strings, wires,

etc. are massless unless

otherwise stated.

Strings and Tension

Consider a heavy box suspended on a rope

attached to a crane.

Does the tension on the rope

increase or decrease if:

(a) the crane accelerates

upward?

(b) the crane accelerates

downward?

(c) the crane moves upward

with constant velocity?

What about a fishing pole?

Pulleys: Demonstration

1. How might a pulley change tension?

2. What would the free-body diagram of the

balance of forces be for a rope and a

pulley:

a. For the rope turned 90 degrees?

b. For the rope turned 180 degrees?

3. Experiment!

Strings, Tension and Pulleys

An ideal pulley is one that simply changes the

direction of the tension.

A man is holding a box

at a constant height off

the ground by means of

a pulley. What are the

forces on the box, the

rope, the pulley, and the

man?

Translational Equilibrium

Objects with zero acceleration are said to be in

equilibrium. For this to happen, the net force on it

must be zero:

This allows the calculation of unknown forces.

Note that the velocity does not need to be zero;

just zero acceleration.

“Translational” refers to motion through space.

Translational Equilibrium

A man is pulling up a bucket at constant velocity.

What are the forces acting on the various parts of

the system?

Translational Equilibrium

What are the forces if the man ties the free end of

the rope also to the bucket?

Translational Equilibrium

What are the forces if the man ties the free end of

the rope also to the bucket?

Example

A 1.84-kg bag of clothespins hangs in the middle

of a clothesline, causing it to sag by an angle of

3.50 degrees. Find the tension in the line.

Demonstration

• Have two people hold a string under

tension. Place a weight in the middle.

How much force must they apply to

straighten out the string?

• In a “cause-and-effect” relationship, the

angle is the “effect”, i.e. the result. What

is(are) the cause(s)?

Two Pulley System

How much force is

required to lift a 52-kg

box at constant

velocity with a two-

pulley system?

Four Pulley System

Static Connected Objects

What are the forces in these two cases?

Pulling Connected Objects

When forces are exerted on connected objects,

their accelerations are the same.

If there are two objects connected by a string,

and we know the force and the masses, we can

find the acceleration and the tension:

Pulling Connected Objects

We treat each box as a separate system:

Pushing Connected Objects:

Contact Forces

Since both objects

have the same

acceleration, F =

(m1+m2)a. However

in the top case

F21=m1a, while in

the bottom case

F12=m2a. Note the

middle box may get

crushed.

Pulling Connected Objects

If there is a pulley, it is easiest to have the

coordinate system follow the string:

Pulleys and …

• Counter-weights

– Atwood Machine

– Elevator

• Springs

– Garage Door Opener

– Compound Bow

Demonstration

• Falling Ball Chain