B. Dynamics · Dynamics: Forces and Motion in a Plane

Connected Objects I: Pulleys, Atwood Machines and the Dumbwaiter

B2.4 B2.5 B2.3About 75 minutes

Learning goals

I can draw a separate free-body diagram for each object in a connected system and write Newton's second law for each.
I can solve the equations together to find the common acceleration and the tensions, including friction.
I can analyse a counterweighted pulley system such as a dumbwaiter or elevator.
I can predict the motion of a pulley system, test the prediction, and explain the difference between prediction and measurement.

Words to know

system
The group of objects being studied together. Forces between them (such as string tension) are internal; forces from outside are external.

ideal string
A string with no mass that does not stretch, so every part of it moves with the same speed and the tension is the same everywhere along it.

ideal pulley
A pulley with no mass and no friction. It changes the direction of the string without changing the tension.

counterweight
A mass on the other side of a pulley that balances most of a load's weight, so a smaller extra force can move the load.

dumbwaiter
A small freight elevator, moved by a rope over a pulley, used to carry food, laundry or goods between floors.

The little elevator

Many older houses, restaurants and libraries have a dumbwaiter: a small box in a shaft that carries dishes or books between floors. In the simplest kind, the box hangs from a rope that runs over a pulley at the top of the shaft to a counterweight, and a person pulls on a rope to move it. Because the counterweight balances most of the load, a gentle pull is enough.

Pulley systems like this are everywhere: elevators, construction cranes, window blinds, exercise machines and flagpoles. In this lesson you will predict exactly how they move.

Remember from Grade 11

In the SPH3U Forces unit you solved toboggan trains, a cart pulled by a hanging mass and the Atwood machine with the two-step method: treat the whole system as one object to find a, then use one object's free-body diagram to find the tension.

The Grade 12 method: one equation per object

The two-step method works for simple systems. For harder systems (friction, angles, three objects) it is safer to do this:

Draw a separate FBD for each object.
Choose a positive direction that follows the string: for example, right for the block on the table and down for the hanging mass. Then every object has the same acceleration a.
Write ΣF = ma for each object along its direction of motion.
Add the equations. The tensions cancel (each appears once as + and once as −), leaving an equation for a. Then substitute back to find each tension.

Adding the equations is exactly the same as the system method, but now you can see where every force goes.

Worked example

Example 1: Block on a rough table

A 3.0 kg block on a table (μk = 0.20) is tied to a 1.5 kg mass hanging over an ideal pulley at the edge. Find the acceleration and the tension.

Block (+ toward the pulley): FN = 3.0 × 9.8 = 29.4 N, so Ff = 0.20 × 29.4 = 5.88 N. Equation: FT − 5.88 = 3.0a.
Hanging mass (+ down): weight = 1.5 × 9.8 = 14.7 N. Equation: 14.7 − FT = 1.5a.
Add the equations: 14.7 − 5.88 = 4.5a, so a = 8.82 ÷ 4.5 = 1.96 m/s².
Substitute: FT = 5.88 + 3.0(1.96) = 11.76 N. Check with the other equation: 14.7 − 11.76 = 2.94 = 1.5 × 1.96 ✓.

a = 1.96 m/s², FT ≈ 11.8 N. The tension is less than the hanging weight (14.7 N), because the hanging mass is accelerating downward.

Left: the set-up. Right: separate free-body diagrams for each object, drawn to scale. The same tension (11.8 N) pulls the block toward the pulley and holds the hanging mass up; the hanging mass's weight is larger than the tension, so it accelerates down.

Worked example

Example 2: The dumbwaiter

A dumbwaiter car with its load has a mass of 40 kg. It hangs from a rope that passes over an ideal pulley to a 32 kg counterweight. The operator pulls down on a rope attached below the counterweight. (a) What pull is needed to raise the car at a constant speed? (b) What pull raises it with an acceleration of 0.50 m/s² upward? (c) What is the tension in the car's rope in (b)?

Car (+ up): T − 40(9.8) = 40a, so T − 392 = 40a.
Counterweight (+ down, following the rope): 32(9.8) + F − T = 32a, so 313.6 + F − T = 32a.
Add: 313.6 + F − 392 = 72a, so F = 78.4 + 72a.
(a) a = 0: F = 78.4 N.
(b) a = 0.50: F = 78.4 + 36 = 114.4 N.
(c) T = 392 + 40(0.50) = 412 N.

(a) 78 N (b) 114 N (c) 412 N. Without the counterweight, lifting the car at constant speed would take 392 N. The counterweight lets the operator use only about a fifth of that.

A counterweighted dumbwaiter. Following the rope, "up" for the car and "down" for the counterweight are the same positive direction, so both share one acceleration a.

Predict, then test

Physics predictions use ideal strings and pulleys. A real pulley has mass (it takes force to spin it up) and friction in its axle, and real strings stretch a little. All of these make the measured acceleration smaller than the ideal prediction. A good investigation (B2.4):

Predicts a from the measured masses and coefficients.
Measures a, for example from the time to travel a known distance from rest: a = 2Δd ÷ Δt².
Compares them with a percent difference, and explains the gap with specific sources of error.
Suggests an improvement (lighter pulley, better axle, longer distance to reduce timing error).

Just for fun

Why are pulleys such good friends?

They always help you take a load off, and they never let a tension go unresolved.

Hands-on activity

Predict and test a table pulley

You need: a toy car (or a small box with a bit of wax paper under it), a kitchen scale, string, a paper cup with a few coins, a smooth thread spool slid onto a pencil (taped at the table edge as a pulley), masking tape, a metre stick and a phone stopwatch.

Measure the mass of the car (m₁) and of the cup with coins (m₂).
Find the car's friction: tilt-test it, or pull it with a luggage scale at a steady speed if it is heavy enough. If friction is very small, ignore it and say so.
Predict: a = (m₂g − Ff) ÷ (m₁ + m₂).
Tie the car to the cup with the string over the spool. Mark a start line and a finish line 0.80 m apart. Release the car from rest and time the run 5 times. Find a = 2(0.80) ÷ Δt² for the average time.
Compute the percent difference. Is the measured value smaller than predicted? List two specific reasons, and one change that would reduce the gap.
Change m₂ (add coins) and repeat. Does the trend match your prediction?

Put a soft towel under the cup so it lands gently.

Your turn

A 4.0 kg block on a frictionless table is tied to a 2.0 kg hanging mass over an ideal pulley. Find the acceleration and the tension.

Repeat the question above with μk = 0.25 between the block and the table.

An Atwood machine has masses of 7.0 kg and 5.0 kg. Find the acceleration and tension.

A 5.0 kg block rests on a table with μs = 0.40. What is the smallest hanging mass that will start it moving?

For the dumbwaiter in Example 2, what pull lowers the car at a constant speed?

A student predicts a = 1.96 m/s² for a pulley system but measures 1.6 m/s². Give two physical reasons the measured value is lower.

More practice: warm-up

In a connected system with a tight string, do all objects have the same size of acceleration?

Is the string tension between two objects in a system internal or external?

What does an ideal pulley change about a string's tension?

If a hanging mass accelerates downward, is the tension larger or smaller than its weight?

Write the formula for finding acceleration from rest using a distance and a time.

More practice: on your own

A 5.0 kg block on a frictionless table is tied on its left to a 3.0 kg hanging mass and on its right to a 1.0 kg hanging mass (each over its own pulley). Find the acceleration and both tensions.

An elevator car (1000 kg) is connected over a pulley to a 900 kg counterweight. What extra force must the motor supply to accelerate the car upward at 1.0 m/s²?

A 6.0 kg block on a table (μk = 0.30) is pulled by a 3.0 kg hanging mass. Starting from rest, how fast is it moving after 0.50 m?

In an Atwood machine with masses m₁ < m₂, explain why the tension must be between m₁g and m₂g.

The predicted acceleration in Example 1 is 1.96 m/s², but a student measures 1.50 m/s². Find the percent difference and the size of the extra resistive force that would explain it.

Challenge

A 70 kg painter sits in a chair hanging from a rope that passes over a pulley fixed above, with the other end of the rope in the painter's hands (a bosun's chair). With what force must the painter pull to rise with an acceleration of 0.40 m/s²?

Derive general formulas for an Atwood machine with masses m₁ < m₂: a = (m₂ − m₁)g ÷ (m₁ + m₂) and FT = 2m₁m₂g ÷ (m₁ + m₂).

In an Atwood machine, a 3.0 kg mass starts 1.0 m above the floor, tied to a 2.0 kg mass resting on the floor. When released, how long does the 3.0 kg mass take to reach the floor, and how high does the 2.0 kg mass rise in total?

Check your understanding

Quiz

A 2.0 kg block on a frictionless table is pulled by a 2.0 kg hanging mass. The acceleration is:

In the system above, the tension in the string is:

When you add the separate Newton's-second-law equations for the objects in a pulley system, the tensions:

The main purpose of a counterweight in a dumbwaiter or elevator is to:

A measured pulley-system acceleration is lower than predicted. Which is the best explanation?

An Atwood machine has 4.0 kg and 6.0 kg masses. The tension is about:

For parents and teachers

Big idea: Connected objects share one acceleration, and each obeys Newton's second law with its own free-body diagram (B2.5). The dumbwaiter and the predict-and-test activity address B2.4: predicting forces in a system, then testing the prediction.

How to help: Ask your teen to point to each object and say which way is positive "along the string". This one habit removes most sign errors.

Common difficulties:

Using the full system mass in one object's equation (or one object's mass for the whole system).
Assuming the tension equals the hanging weight when the system accelerates.
Forgetting friction on the table, or using μs for a block that is already moving.
Expecting the measured acceleration to match the ideal prediction exactly.

Safety: use light masses (a cup of coins), and pad the floor where they land.