D. Structures and Mechanisms · Systems in Action

Input Force, Output Force and Mechanical Advantage: Levers

D2.6 D2.4 A1.2About 45 minutes

Learning goals

I can explain the relationship between the input force and the output force of a machine.
I can calculate mechanical advantage from forces, and ideal mechanical advantage of a lever from its arm lengths.
I can identify first-, second- and third-class levers.
I can explain the trade-off between force and distance.

Words to know

input force
The force you put into a machine. Also called the effort force.

output force
The force the machine puts on the load. Also called the load force.

mechanical advantage (MA)
How many times a machine multiplies the input force. MA = output force ÷ input force.

ideal mechanical advantage (IMA)
The mechanical advantage a machine would have with no friction, found from distances. For a lever: input arm ÷ output arm.

fulcrum
The fixed point a lever turns on.

input arm and output arm
On a lever, the distance from the fulcrum to the input force, and from the fulcrum to the load. Also called the effort arm and load arm.

Hook: the stuck paint can

In Grade 4 you saw a screwdriver used as a lever to pop open a paint can lid. Your hand pushes with a small force, and the screwdriver's tip pushes the lid with a much bigger force. How much bigger? In Grade 8 you will answer that with a number: the mechanical advantage.

A screwdriver's tip is wedged under the rim of a metal paint can lid on a workbench. — The rim of the can is the fulcrum. A small push down on the long handle makes a much bigger force at the short tip to pop the lid. AI-generated image

Input force, output force and mechanical advantage

Every machine has:

an input force (effort): the force you apply
an output force (load force): the force the machine applies to the load

The mechanical advantage (MA) compares them:

MA = output force ÷ input force

MA has no units because newtons divide by newtons.

MA greater than 1: the machine multiplies force. Example: a pry bar, MA = 8 means the output force is 8 times your input.
MA equal to 1: same force, but it may change direction. Example: a single fixed pulley.
MA less than 1: the output force is smaller than your input, but the output moves farther and faster. Example: a hockey stick or a fishing rod.

The trade-off: force for distance

Machines never give you something for nothing. If a machine multiplies your force by 4, you must move 4 times as far as the load moves (ideally). In the ideal case, the work in equals the work out:

input force × input distance = output force × output distance

In real machines, some work is always lost to friction, so the real (actual) MA is a little less than the ideal MA.

The three classes of levers. F is the fulcrum (triangle), L is the load (box) and E is the input force or effort (arrow).

Class	What is in the middle	MA	Examples
First class	the fulcrum	can be more than, equal to or less than 1	seesaw, pry bar, scissors, a screwdriver opening a paint can
Second class	the load	always more than 1	wheelbarrow, nutcracker, bottle opener
Third class	the input force	always less than 1	tweezers, broom, hockey stick, fishing rod, your forearm

The three classes of levers.

Ideal mechanical advantage of a lever

For a lever, measure from the fulcrum:

input arm: fulcrum to where you push
output arm: fulcrum to the load

IMA = input arm ÷ output arm

A lever balances when: input force × input arm = output force × output arm. That is why a long handle helps so much.

Worked example

Example 1: MA from forces

A pry bar lifts a 400 N rock when a person pushes down with 50 N. What is its mechanical advantage?

MA = output force ÷ input force
MA = 400 N ÷ 50 N = 8
The pry bar multiplies the input force 8 times.

MA = 8

Worked example

Example 2: A wheelbarrow (second-class lever)

In a wheelbarrow, the wheel's axle is the fulcrum. A 600 N load of soil sits 0.4 m from the axle. Your hands lift the handles 1.2 m from the axle. What is the IMA, and what lifting force is needed (ignoring friction)?

IMA = input arm ÷ output arm = 1.2 m ÷ 0.4 m = 3.
Input force = output force ÷ IMA = 600 N ÷ 3 = 200 N.
Check the balance: 200 N × 1.2 m = 240 and 600 N × 0.4 m = 240. ✓

IMA = 3, so you lift with about 200 N to hold up 600 N of soil.

The wheelbarrow in Example 2 is a second-class lever: the load sits between the fulcrum (the wheel axle) and your hands. The input arm is 3 times the output arm, so a 200 N lift holds up 600 N of soil.

Worked example

Example 3: A third-class lever trades force for speed

A person casting a fishing rod puts in 40 N with their upper hand (the hand at the end of the handle is the fulcrum), and the tip of the rod pushes on the line with 10 N. Find the MA. Why would anyone use this lever?

MA = 10 N ÷ 40 N = 0.25.
The output force is only a quarter of the input force.
But the tip moves 4 times as far and as fast as the hand (ideally). That speed flings the lure far out over the lake.

MA = 0.25. A third-class lever trades force for distance and speed.

Just for fun

The ancient Greek scientist Archimedes is said to have boasted, "Give me a place to stand and a lever long enough, and I will move the Earth." Bold words from someone who never found a fulcrum big enough to test it.

Hands-on activity

Ruler-and-coin lever lab

You need: a 30 cm ruler, a pencil or marker to use as a fulcrum, 6 identical coins (all the same kind), tape (optional).

Safety: keep coins away from young children, who could choke on them.

Put the pencil under the ruler at the 15 cm mark so the empty ruler balances.
Stack 2 coins centred on the 9 cm mark (6 cm from the fulcrum). Where must 1 coin go on the other side to balance? Predict, then test. (Prediction: 2 × 6 = 1 × ?, so 12 cm from the fulcrum, the 27 cm mark.)
Now put 3 coins on the 11 cm mark (4 cm from the fulcrum). Predict and test where 1 coin balances them.
Record your results in a table: coins on left, distance left (cm), coins on right, distance right (cm), coins × distance on each side.
Think of the single coin as your input force and the stack as the load. What is the ideal MA in each test?

What to look for: in test 2 the single coin balances 12 cm from the fulcrum (27 cm mark); in test 3 it is also 12 cm away, because 3 × 4 = 12. The IMA is 12 ÷ 6 = 2 in test 2 and 12 ÷ 4 = 3 in test 3. Results within about 1 cm are good: real rulers and coins are not perfect (A1.2).

Your turn

Write the formula for mechanical advantage.

A bottle opener puts 100 N on a cap when you push with 25 N. Find the MA.

What does an MA of 1 mean?

Classify each lever: a) seesaw b) wheelbarrow c) tweezers.

A lever has an input arm of 90 cm and an output arm of 15 cm. What is its IMA?

Why would anyone use a machine with an MA less than 1?

A 200 N load is 0.5 m from a lever's fulcrum. You push 2 m from the fulcrum on the other side. What input force balances it (ignoring friction)?

More practice: warm-up

What is the fulcrum of a lever?

Output force 600 N, input force 200 N. What is the MA?

In which class of lever is the load in the middle?

Which class of lever always has an MA less than 1?

Does an MA of 5 mean the machine multiplies or reduces your force?

More practice: on your own

A nutcracker's hinge is the fulcrum. Your hand squeezes 15 cm from the hinge and the nut is 3 cm from the hinge. Find the IMA and the ideal force on the nut if you squeeze with 20 N. What class of lever is it?

Maya (350 N) sits 1.6 m from the middle of a seesaw. How far from the middle must Leo (400 N) sit on the other side to balance?

When you sweep, your top hand stays still (the fulcrum), your lower hand pushes 0.4 m below it, and the bristles are 1.2 m below your top hand. Find the IMA and explain the benefit.

A lever has an IMA of 4, but when tested it lifts 350 N with 100 N of input force. Find the actual MA and explain the difference.

A lever has an IMA of 4. To lift a load 10 cm, how far must you push the input end? Show that the work in equals the work out for a 600 N load.

Challenge

You need to lift a 900 N stone with a pry bar 1.4 m long, and you can push with at most 150 N. Where should you put the fulcrum (rock under the bar)? Explain.

Your forearm is a third-class lever: the elbow is the fulcrum. Suppose your biceps muscle attaches 4 cm from the elbow and your hand holds a 20 N ball 32 cm from the elbow. What force must the biceps pull with? What is the MA, and why is this design useful?

Scissors are first-class levers. Explain why it is easier to cut thick cardboard near the hinge than near the tips.

Check your understanding

Quiz

Mechanical advantage is found by...

You push with 30 N and a lever lifts 150 N. What is the MA?

Which is a third-class lever?

An MA less than 1 means the machine...

A lever has an input arm of 2 m and an output arm of 0.5 m. What is its IMA?

Why is a real lever's MA a bit less than its ideal MA?

For parents and teachers

Big idea (D2.6): Students explain the relationship between input and output forces and determine the mechanical advantage of various mechanical systems. Two formulas: MA = output force ÷ input force (from measured forces) and, for levers, IMA = input arm ÷ output arm (from distances). Lever classes extend the levers your child met in Grades 2 and 4.

How to help: Hunt for levers at home (scissors, bottle opener, tongs, nail clippers, a broom) and classify them by what is in the middle. The coin lab is a good fair-test experiment (A1.2): keep the coins identical and the fulcrum fixed.

Common difficulties:

Dividing the wrong way. Output (what the machine gives) goes on top.
Thinking machines create energy. Remind your child of the trade-off: more force means more distance.
Measuring arms from the end of the lever instead of from the fulcrum.