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Two forces on one line, the resultant they leave, and why balanced forces do not mean a standstill.
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.
You will work out the resultant of forces acting along one line, say whether they are balanced, and predict what happens to the object's motion — including the case where the forces are balanced and it is still moving.
From the last lesson: every force is a push or a pull between two things, and it is measured in newtons. Now there is more than one force on the same object at once. You can also already add and subtract whole numbers, which is all the arithmetic this needs.
You have felt this lesson already, too. In a tug of war, when both sides pull equally hard, the rope does not move. When one side pulls harder, the rope slides their way. That is balanced and unbalanced forces.
| Term | What it means |
|---|---|
| Resultant | What all the forces on a thing add up to, once they have canceled each other as far as they can. |
| Balanced | The forces cancel exactly: the resultant is zero. |
| Unbalanced | The forces do not cancel: there is a resultant. |
| Newton (N) | The unit a force is measured in. |
| Direction | Which way a force points: left, right, up or down. |
| Constant speed | Moving at a speed that does not change. |
When two forces act on one thing along one line, they pull against each other. Take the smaller from the bigger and what is left is the resultant.
$$7\text{ N right} \quad\text{against}\quad 4\text{ N left} \;=\; 3\text{ N right}$$
If the resultant is zero, the forces are balanced, and the rule is: nothing about how the object is moving changes. If it was still, it stays still. If it was gliding along, it keeps gliding along at the same speed in the same direction.
If the resultant is not zero, the forces are unbalanced, and the object speeds up in the direction the resultant points. If it was already moving the other way, it slows down first.
Forces that point the same way help each other, so they add. Forces that point opposite ways fight each other, so they subtract.
Another way: action
Two people pull a rope, one at each end, as hard as each other. The rope does not move. Now one of them eases off a little: the rope goes their opponent's way at once. Nothing changed except which force was bigger.
Another way: steps
Every question of this kind is the same three lines.
| Forces | Resultant | What happens |
|---|---|---|
| 6 N left, 6 N right | 0 N | its motion does not change |
| 4 N left, 9 N right | 5 N right | it speeds up to the right |
| 10 N left, 3 N right | 7 N left | it speeds up to the left |
Notice the middle column is never a total. Forces that pull opposite ways subtract. Adding them is the most common slip, and it gives an answer bigger than either force, which is a thing that cannot happen when two forces are fighting each other.
Sometimes more than two forces act at once. Two children might pull a sled together, both to the right, while the snow rubs to the left. Then there are two jobs, and the order matters.
First, add the forces that point the same way. Two pulls of $8$ N and $6$ N to the right make $14$ N to the right: they are on the same side, so they help each other. Second, subtract across the two sides. Against $5$ N of rubbing to the left, the resultant is $14 - 5 = 9$ N to the right.
A good way to keep track is to draw the object as a box and each force as an arrow, with its number written on it. Scientists call this a force diagram. Longer arrows stand for bigger forces, so you can often see which side wins before you do the arithmetic.
Think about a whole trip on a bicycle, and what the forces are doing at each stage.
| Stage | Forces along the road | What happens |
|---|---|---|
| pushing off | pedaling beats the rubbing | speeds up |
| cruising | pedaling equals the rubbing | steady speed |
| braking | the brakes and rubbing win | slows down |
| stopped | nothing pushes along the road | stays still |
Two of those rows are balanced, and only one of them is standing still. While cruising, the cyclist is moving fast with balanced forces. That is not a trick: it is exactly what balanced means. The forces decide whether the speed changes, not whether there is any speed.
Every resultant should pass three checks.
The first check catches the most common mistake. The second catches the next most common one: getting the size right but saying the thing speeds up the wrong way.
A force diagram is a quick sketch that makes the arithmetic almost automatic. Draw the object as a simple box. Then, for every force on it, draw an arrow starting at the box and pointing the way the force acts. Write the size in newtons beside each arrow.
Make the arrows roughly as long as the forces are big: a $10$ N arrow twice as long as a $5$ N arrow. Now the diagram does some of the thinking for you. If the arrows on the left and right are the same length, the forces are balanced. If one side's arrows are longer, that side wins, and the resultant points that way.
Label each arrow with the thing that makes the force: rope, snow, hand, Earth. That keeps the rule from the first lesson in view: every force on the diagram has a second thing at its other end, even though only the object itself is drawn.
Forces act up and down too. A book resting on a table is pulled down by the Earth, which is its weight. The table pushes it up just as hard. The two forces are balanced, which is why the book does not sink into the table or jump off it.
An elevator shows both cases. While it waits at a floor, the cable pulls up as hard as the Earth pulls down. As it starts to go up, the cable pulls harder than the weight, and there is a resultant upward, so it speeds up. Once it is moving at a steady speed between floors, the cable's pull and the weight are balanced again, even though the elevator is moving. As it slows to stop at the top, the cable pulls a little less than the weight, and the resultant points down.
Up-and-down forces follow exactly the same rule as left-and-right ones. Only the directions have new names.
On field day, two classes face each other across a line with a long rope. The red team has four students pulling with about $150$ N, $140$ N, $130$ N and $120$ N. Their forces all point the same way, so they add: $540$ N. The blue team's four pull with $160$ N, $130$ N, $120$ N and $110$ N, which adds up to $520$ N.
The resultant is $540 - 520 = 20$ N toward the red team, so the rope starts to slide their way. It is a small resultant, which is why the rope creeps rather than jumps. If one blue student slips and lets go, the blue total drops to about $400$ N, the resultant jumps to $140$ N, and the rope races toward the red side.
When both teams pull equally hard, the rope does not move at all, even though eight students are pulling as hard as they can. Balanced forces can be very large and still cancel out completely.
A car driving along a highway at a steady $65$ miles per hour has balanced forces on it along the road. Its engine turns the wheels, and the road pushes the car forward with a force of about $600$ N. The air pushes back on the front of the car, and the tires rub on the road; together they push backward with about $600$ N too. The resultant is $600 - 600 = 0$ N, so the speed does not change.
When the driver presses the gas pedal, the forward push grows to perhaps $1500$ N. Now $1500 - 600 = 900$ N points forward, and the car speeds up. When the driver brakes, the backward forces become much bigger than the forward push, and the car slows down.
Engineers who design cars study these forces closely. A smoother body shape means the air pushes back less, so the engine needs less forward push to keep the forces balanced at highway speed, and the car uses less fuel.
Almost everyone first reads balanced forces as the thing is not moving. It is easy to see why: books on tables are the example everyone is shown, and books on tables sit still.
But a skydiver falling steadily has balanced forces on her, and she is going down fast. A train at a constant speed on a straight track has balanced forces on it. Balanced forces guarantee no change — and carrying on at the same speed is exactly no change.
The other half of the same mistake is believing that moving needs a force. It does not. Moving needs nothing. Changing how you move is the thing that needs a resultant force, and that is the only job a resultant force has.
A third slip is adding forces that point opposite ways. Someone who sees $9$ N right and $4$ N left and writes $13$ N has found a force bigger than either one, which is impossible when the two are fighting each other. Check the directions before you choose to add or subtract.
List the forces on the box.
$8\ \text{N left}, \quad 8\ \text{N right}$
Two forces, pulling opposite ways.
Check for forces the same way.
$\text{none to add}$
One force on each side.
Take the smaller from the bigger.
$8 - 8 = 0\ \text{N}$
Equal and opposite forces cancel completely.
Name the result.
$\text{balanced}$
The resultant is zero.
Predict what happens.
$\text{no change in motion}$
The box was still, so it stays still. If it had been sliding, it would keep sliding at the same speed.
List the forces on the sled.
$9\ \text{N right (rope)}, \quad 4\ \text{N left (snow)}$
The rope pulls one way and the snow rubs the other.
Check for forces the same way.
$\text{none to add}$
One force on each side again.
Take the smaller from the bigger.
$9 - 4 = 5\ \text{N}$
Opposite forces subtract.
Give the direction.
$5\ \text{N to the right}$
The way the bigger force, the rope, points.
Check against the biggest side.
$5 < 9$
A resultant is never bigger than the bigger side.
Predict what happens.
$\text{speeds up to the right}$
An unbalanced force changes the motion.
List the forces on the couch.
$\text{pushes: } 60\ \text{N and } 50\ \text{N right; rubbing: } 80\ \text{N left}$
Three forces this time.
Add the pushes that point the same way.
$60 + 50 = 110\ \text{N right}$
The two movers help each other.
Compare the two sides.
$110\ \text{N right against } 80\ \text{N left}$
Now there is one total each way.
Take the smaller from the bigger.
$110 - 80 = 30\ \text{N}$
Opposite totals subtract.
Give the direction.
$30\ \text{N to the right}$
The way the movers push.
Predict what happens.
$\text{the couch speeds up to the right}$
The forces are unbalanced.
Check with one mover alone.
$60 < 80: \text{ it would not start}$
One mover could not beat the rubbing, which is why it took two.
Take the smaller from the bigger.
$12 - 5 = 7\ \text{N}$
Opposite forces subtract.
Give the direction.
Predict what happens.
Two forces act on a rope in a tug of war along one line: $15$ N pulling left and $15$ N pulling right. What is the size of the resultant force, in newtons?
Answer: unit: N / kN
Complete the worked solution: in a tug of war, the left team's two players pull with $37$ N and $21$ N. The right team's two players pull with $21$ N and $32$ N. What is the resultant force on the rope?
Add the left team's pulls.
$37 + 21 =$ l N
They pull the same way, so they add.
Add the right team's pulls.
$21 + 32 =$ r N
They pull the same way too.
Take the smaller from the bigger.
$\text{left} - \text{right} =$ s N, to the left
The two teams pull opposite ways, and the left team is stronger.
On a cart in a hallway, $10$ N pulls left and $3$ N pulls right. What happens to how it is moving?
A sled is standing on snow and someone starts to pull. Put these four moments in the order they happen.
Number the steps in order (write the number in the box):
Three boxes, each pulled from both sides. Work out the resultant force on each, in newtons.
| pull to the left (N) | pull to the right (N) | resultant force (N) | |
|---|---|---|---|
| pulled equally from both sides | 5 | 5 | |
| pulled harder from the left | 6 | 4 | |
| pulled harder from the right | 5 | 9 |
Four forces act on this sled. Click the two that act along the snow, the pair that decides whether it speeds up or slows down.
This task has no paper form; do it on a device.
On a box in a tug of war the two forces are $6$ N and $6$ N, pulling opposite ways. Mark the size of the resultant on the scale.
0 |——————————| 20
Mark the position with a cross, then write the value:
Two children pull a sled to the right with ropes: one pulls with $14$ N and the other with $7$ N. The snow rubs on the sled with $7$ N to the left. What is the size of the resultant force on the sled, in newtons?
Answer: unit: N / kN
A skydiver's parachute has been open for a while and her speed has stopped changing. She is still going down, steadily. What are the forces on her doing?
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Three boxes, each pulled from both sides. Work out the resultant force on each, in newtons.
| pull to the left (N) | pull to the right (N) | resultant force (N) | |
|---|---|---|---|
| pulled equally from both sides | 3 | 3 | |
| pulled harder from the left | 10 | 4 | |
| pulled harder from the right | 2 | 9 |
You can find a resultant and say what it does to the motion. Tell someone why a skydiver falling at a steady speed has balanced forces on her.
17. Your turn: 12 N pulls left, 5 N pulls right. Resultant, and then what happens?, step 2
$7\ \text{N to the left}$
The bigger force points left.
17. Your turn: 12 N pulls left, 5 N pulls right. Resultant, and then what happens?, step 3
$\text{speeds up to the left}$
Unbalanced forces change the motion.