Back to the on-screen lesson ·
A force's turning effect is force times distance from the pivot; an object balances when the turning effects on each side are equal.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to find a torque, and the missing force or distance that balances a seesaw or lever.
You can find forces in newtons, including weights, and you know what it means for forces to balance. This lesson looks at forces that turn things rather than push them along: doors, wrenches, seesaws and levers.
| Term | What it means |
|---|---|
| Torque | The turning effect of a force, $\tau = Fd$, also called its moment. |
| Pivot | The point an object turns about, such as a hinge or axle. |
| Newton meter | The unit of torque, N·m. |
| Lever | A bar that turns about a pivot to multiply a force. |
| Balance | When the turning effects on each side of a pivot are equal. |
| Mechanical advantage | How many times a lever multiplies the force put in. |
The turning effect of a force about a pivot is its torque:
$$\tau = Fd,$$
where $F$ is the force and $d$ its distance from the pivot, measured at right angles to the force.
Another way: picture
Picture opening a heavy door. Pushing near the handle, far from the hinges, it swings easily. Pushing near the hinges, you strain and it barely moves. The push is the same, but its distance from the hinges, and so its turning effect, is very different.
Another way: steps
Newton meters means a force in newtons multiplied by a distance in meters. A $40$ N push on a wrench $0.3$ m from the bolt gives a torque of $40 \times 0.3 = 12$ N·m.
Unlike density and pressure, which divide, torque multiplies. The unit, newtons times meters, says so. Both a larger force and a longer distance give a larger turning effect.
The distance in torque is always measured from the pivot, the point the object turns about. For a door, that is the hinges; for a wrench, the bolt; for a seesaw, the center support.
A force right at the pivot has zero distance and no turning effect at all. Pushing on a door right at its hinges does nothing, however hard you push.
A seesaw balances when the turning effects on its two sides are equal. A child weighing $300$ N sitting $2$ m from the pivot makes a turning effect of $600$ N·m. To balance, a child weighing $400$ N must sit $1.5$ m from the pivot on the other side, since $400 \times 1.5 = 600$.
The heavier child sits closer to the pivot. Equal weights are not needed for balance; equal turning effects are.
A lever is a bar that turns about a pivot. Put a heavy load close to the pivot and push far from it on the other side, and a small push balances a large load. A crowbar can lift a rock many times heavier than the push on its handle.
The trade-off is distance. The long end of the lever moves much farther than the short end. You push less hard, but you push through a greater distance.
A longer wrench gives more turning effect for the same push, which is why stuck bolts are loosened with long wrenches or breaker bars. Door handles are placed far from the hinges for the same reason.
Screwdrivers with thick handles, faucet handles and steering wheels all use distance to turn a modest force into a useful turning effect.
Checking an answer. A heavier load must sit closer to the pivot to balance a lighter one. A longer handle must need less force. The unit must be newton meters.
Experiments show that what decides whether an object turns is not the force alone but force times distance. Double the distance and the same force turns things twice as strongly.
When the turning effects on both sides are equal, they cancel, and the object does not start to turn. That is why setting them equal finds the balance point, the same way balanced forces find an object at rest.
Your arm is a lever. The elbow is the pivot, and the biceps muscle pulls on the forearm just a few centimeters from the elbow. A book in your hand is about thirty centimeters away.
Because the muscle is so close to the pivot, it must pull with many times the book's weight to hold it up. Our bodies trade force for speed and range: a small muscle movement swings the hand a long way.
Scissors, pliers, bottle openers and nutcrackers are all levers. In each, a pivot sits between or beside two arms, and a small force on the long part makes a large force on the short part.
Wheelbarrows are levers too. The wheel is the pivot, the load sits close to it, and the handles are far away, so lifting the handles takes much less force than lifting the load directly.
A hanging mobile is a set of balanced levers. Each rod hangs from a string that acts as its pivot, with heavier shapes hung closer to the string and lighter ones farther out.
The American artist Alexander Calder became famous for mobiles in which every rod balances perfectly, so the whole sculpture turns gently in the slightest breeze.
The most common error is using the length of the whole bar instead of the distance from the pivot. Another is to divide force by distance, as in pressure, instead of multiplying.
A third is thinking a balanced seesaw needs equal weights. It needs equal turning effects, so the heavier child moves closer to the center.
Turning effects have a direction: clockwise or counterclockwise. On a seesaw, the child on the left turns it one way and the child on the right turns it the other.
For balance, the total clockwise turning effect must equal the total counterclockwise one. With several children on each side, add the turning effects on each side first, then compare.
Opening a door takes a turning effect of a few newton meters. Loosening a car's lug nut takes about a hundred. A person's weight on a pedal of a bicycle gives several hundred.
These benchmarks help check answers. A calculation that says a door needs thousands of newton meters probably used centimeters where meters were needed.
In some countries, the turning effect of a force is called its moment. American textbooks usually call it torque. Both words mean the same thing: force times distance from the pivot, in newton meters.
Engineers and mechanics use torque every day, from tightening bolts to designing engines, whose power reaches the wheels as torque on the axles.
The distance in a torque is measured at right angles to the force. When you push straight across a wrench handle, that distance is simply how far along the handle you push. When you push at a slant, only part of your push turns the wrench, and the turning effect is smaller.
That is why mechanics try to push or pull at right angles to the handle. A push that points along the handle, toward or away from the bolt, gives no turning effect at all, no matter how hard it is. In this course, forces act at right angles, and the distance is measured straight along the bar.
More than two thousand years ago, the Greek scientist Archimedes studied levers and discovered the balance rule: weights balance when their distances from the pivot are in the opposite ratio to the weights. He is said to have boasted that, given a place to stand and a long enough lever, he could move the whole Earth.
He was exaggerating, since the lever would have to be impossibly long, but the idea behind the boast is exactly this lesson's: a small force far from the pivot can balance an enormous force close to it.
Construction cranes lift heavy loads on a long arm. To keep the crane from tipping over, a heavy counterweight sits on a short arm on the other side of the tower. The counterweight's turning effect balances the load's, so the tower stays upright.
Crane operators check a load chart before every lift. The farther out the load, the larger its turning effect, so the crane can lift less weight far from the tower than close to it.
To check a balance answer, multiply each weight by its distance and compare the two products. If they match, the seesaw balances; if not, the side with the larger product goes down.
Seesaws in parks across the country are classic balancing machines. When two children of different weights play, the heavier child scoots toward the middle until the seesaw balances, which happens when both turning effects are equal.
Some seesaws have several seats at different distances, so children can balance without moving along the board. Playground designers also place the pivot at the exact center of a uniform board, so the board's own weight balances and does not tip toward either side. Once balanced, a small push with the legs is enough to set the seesaw rocking up and down.
A wheelbarrow is a lever with the wheel as the pivot. The load sits in a tub close to the wheel, and the handles stick out far behind. Lifting the handles, the gardener supplies a small force at a large distance, balancing a heavy load at a small distance.
A load of wet soil weighing seven hundred newtons might need a lift of only two hundred newtons at the handles. Gardeners learn to pile the load toward the front of the tub, close to the wheel, because that shortens the load's distance and makes lifting even easier. Loading it toward the back makes the barrow feel much heavier.
It is natural to think a seesaw balances only when the children weigh the same. But what must match is the turning effect, weight times distance. A heavy child close to the pivot balances a light child far away.
A related error is to think a longer wrench makes a bolt easier to turn because it is somehow stronger. It gives the same push a longer distance from the bolt, and so a bigger turning effect.
A $20$ N push acts at right angles on a door, $0.8$ m from the hinges. Find the torque.
$\tau = 20 \times 0.8 = 16\ \text{N·m}$
Force times distance.
Find the torque pushing $0.2$ m from the hinges.
$\tau = 20 \times 0.2 = 4\ \text{N·m}$
A quarter as much.
Find the push needed there for $16$ N·m.
$F = \dfrac{16}{0.2} = 80\ \text{N}$
Four times the push.
Find the torque pushing at the hinges.
$\tau = 20 \times 0 = 0$
No turning.
Explain handle placement.
$\text{far from the hinges}$
Largest distance.
Children weighing $250$ N at $2$ m and $200$ N at $1$ m sit on the left. Find the left turning effect.
$250 \times 2 + 200 \times 1 = 700\ \text{N·m}$
Add both.
A $350$ N child sits on the right. Write the balance rule.
$350 \times d = 700$
Equal turning effects.
Solve for the distance.
$d = \dfrac{700}{350} = 2\ \text{m}$
On the right.
Check the right turning effect.
$350 \times 2 = 700\ \text{N·m}$
Balanced.
Predict what happens if the right child moves to $1.5$ m.
$525 < 700 \Rightarrow \text{left side goes down}$
Unbalanced.
Find the extra weight needed at $1.5$ m.
$\dfrac{700}{1.5} - 350 = 116.7\ \text{N}$
A backpack would do.
A crowbar's pivot is $0.1$ m from an $800$ N crate edge. Find the crate's turning effect.
$\tau = 800 \times 0.1 = 80\ \text{N·m}$
Weight times distance.
The worker pushes $1.0$ m from the pivot. Write the balance rule.
$F \times 1.0 = 80$
Equal turning effects.
Solve for the push.
$F = 80\ \text{N}$
A tenth of the load.
Find the force multiplication.
$\dfrac{800}{80} = 10$
The distance ratio.
The worker pushes the end down $0.3$ m. Find how far the crate rises.
$0.3 \div 10 = 0.03\ \text{m}$
Ten times less.
Find the push with a $2.0$ m bar.
$F = \dfrac{80}{2.0} = 40\ \text{N}$
Longer, easier.
Explain the trade-off.
$\text{less force, more distance}$
Levers save force, not work.
Write the formula.
$\tau = Fd$
Force times distance.
Substitute the values.
$\tau = 50 \times 0.4$
Newtons times meters.
Evaluate the torque.
A mechanic pushes at right angles on a wrench with $40$ N, $0.3$ m from the bolt. What torque does the push give, in N·m?
Complete the worked solution: a gardener uses a steel bar as a lever to lift a $600$ N rock. The rock sits $0.2$ m from the pivot, and the gardener pushes down $1.2$ m from it on the other side. Find the rock's turning effect in N·m, the push needed in N, and how many times the push is multiplied.
Find the rock's turning effect.
$\tau = W \times d_{\text{rock}} =$ t
Weight times distance.
Find the push needed.
$F = \dfrac{\tau}{d_{\text{push}}} =$ f
Equal turning effects.
Find how many times the push is multiplied.
$\dfrac{W}{F} =$ r
The ratio of the distances.
Note the trade-off.
$\text{the long end moves farther}$
Less force, more distance.
Match each idea to its statement.
| force times distance from the pivot | equal turning effects on each side | the same torque with less force | the point an object turns about | |
|---|---|---|---|---|
| torque | ||||
| a balanced seesaw | ||||
| a longer wrench | ||||
| the pivot |
On a playground seesaw, a child weighing $360$ N sits $1.5$ m from the pivot. Fill in that child's turning effect in N·m, how far from the pivot in m a child weighing $450$ N must sit on the other side to balance, and the downward force in N that would balance it from $3$ m instead.
| value | |
|---|---|
| first child's turning effect (N·m) | |
| balancing distance (m) | |
| balancing force at 3 m (N) |
A student pushes a door at right angles with a steady $20$ N, at a distance $d$ from the hinges, in meters. Write the torque about the hinges, in N·m, as a function of $d$.
Answer:
A bolt needs a torque of $30$ N·m to loosen. Using a wrench that lets you push at right angles $0.25$ m from the bolt, what force is needed, in N?
Answer: N
At a community garden, a volunteer loads a wheelbarrow with $1000$ N of sand. The load's weight acts $0.4$ m from the wheel's axle, and the handles are $1.6$ m from it. What upward force on the handles lifts the load, in N?
Answer: N
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A student pushes a door at right angles with a steady $12.5$ N, at a distance $d$ from the hinges, in meters. Write the torque about the hinges, in N·m, as a function of $d$.
Answer:
You can use torque and balance. Explain to someone why a heavy child must sit closer to the middle of a seesaw to balance a lighter one.
29. Your turn: a $50$ N force acts $0.4$ m from a pivot. What is its torque?, step 3
$\tau = 20\ \text{N·m}$
Newton meters.