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A net force makes an object accelerate by $a = F/m$: more force, more acceleration; more mass, less.
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By the end of this lesson you will be able to use $F = ma$ to find an acceleration, a force or a mass from the other two.
You can find an acceleration from a change in velocity, and combine forces into a net force. This lesson connects the two: the net force on an object decides its acceleration, through its mass.
| Term | What it means |
|---|---|
| Newton's second law | The net force equals mass times acceleration, $F = ma$. |
| Mass | How much matter an object has, in kilograms; it resists acceleration. |
| Net force | The combined force on an object, after opposite forces are subtracted. |
| Acceleration | The change in velocity each second, in m/s². |
| Newton | The force that gives $1$ kg an acceleration of $1$ m/s². |
| Inertia | An object's resistance to changes in its motion, measured by its mass. |
Newton's second law links net force, mass and acceleration:
$$F = ma, \qquad a = \dfrac{F}{m}.$$
One newton is exactly the net force that gives one kilogram an acceleration of one meter per second squared.
Another way: picture
Picture pushing an empty shopping cart and then a full one with the same effort. The empty cart speeds up quickly; the full one barely gets going. The push is the same, but the mass is different, and more mass means less acceleration.
Another way: steps
For a given object, doubling the net force doubles the acceleration. Pushing a sled twice as hard makes its speed climb twice as fast. The acceleration is proportional to the net force.
Zero net force means zero acceleration: the velocity stays the same. That fits what you learned about balanced forces. Newton's second law adds the numbers: how much acceleration each newton of unbalanced force produces.
For a given net force, doubling the mass halves the acceleration. A loaded truck speeds up more slowly than an empty one with the same engine force, because it has more mass to move.
Mass measures inertia, an object's resistance to changes in its motion. The more mass, the more force it takes to speed the object up, slow it down or turn it.
The law can be used three ways. To find acceleration, divide force by mass. To find the force needed, multiply mass by acceleration. To find an unknown mass, divide force by acceleration.
A $40$ N net force on an $8$ kg cart gives $5$ m/s². To give a $50$ kg sled $2$ m/s² takes $100$ N of net force. A box that accelerates at $3$ m/s² under $36$ N has a mass of $12$ kg.
The F in Newton's second law is the net force, found after opposite forces are subtracted. Pushing a crate with $100$ N against $40$ N of friction gives a net force of $60$ N, and only that $60$ N produces acceleration.
Using the push alone would overestimate the acceleration. Always find the net force first, from a free-body diagram, and then apply the law.
The unit of force is defined by this law. One newton is the net force that gives a one-kilogram mass an acceleration of one meter per second squared. So a newton is a kilogram meter per second squared.
This is why dividing newtons by kilograms gives meters per second squared: the units work out automatically. A small apple weighs about one newton, so the unit is easy to picture.
Checking an answer. More mass must give less acceleration for the same force. The units must come out as m/s² or N. The net force, not the push alone, must be used.
Newton's second law is a law of nature found by experiment: carefully measured accelerations are always proportional to the net force and inversely proportional to the mass. Countless experiments since Newton's time have confirmed it.
Because the law involves the net force, all the ways of combining forces from the last lesson apply directly. Find the net force correctly, and the law does the rest.
Weight is the force of gravity, and Newton's second law explains why it equals $mg$. A falling object with only gravity acting accelerates at $g = 9.8$ m/s², so the force on it must be its mass times $9.8$.
A heavier object has more weight but also more mass, so every object falls with the same acceleration if air resistance is small. A hammer and a feather dropped on the Moon, where there is no air, land together.
In sports, athletes use the second law without thinking about it. A pitcher throws a light baseball faster than a shot-putter throws a heavy iron ball, because the same arm force gives the lighter ball a larger acceleration.
Football linemen use their mass to resist being pushed back. A heavier player accelerates less when an opponent pushes, which is why offensive lines are made of the heaviest players on the field.
A car's engine can supply only so much force, so a car loaded with passengers and luggage accelerates more slowly than an empty one. Trucks carrying heavy loads take a long time to get up to highway speed.
The same law explains braking. A heavily loaded truck needs a larger braking force to slow down at the same rate as a car, which is why trucks have bigger brakes and need more room to stop.
The most common error is to use the applied push instead of the net force, forgetting friction. Another is to multiply force by mass when finding acceleration, instead of dividing.
Mixed units cause trouble too: a mass in grams or metric tons must be converted to kilograms, and a force in kilonewtons to newtons, before using the law.
A person pushing hard can exert a few hundred newtons. On a $20$ kg cart with little friction, that is an acceleration of more than ten meters per second squared for a moment. On a $1000$ kg car, the same push gives only a few tenths of a meter per second squared, which is why pushing a car takes several people.
Comparing the force to the mass tells you quickly whether an answer is reasonable.
Because $m = F/a$, a known force and a measured acceleration reveal an object's mass. This works even where weighing does not, such as in orbit, where everything floats.
Astronauts on the International Space Station measure their mass with a device that applies a known force and measures their acceleration. The second law turns those two numbers into a mass, with no scale needed.
The acceleration always points the same way as the net force. If the net force points backward on a moving car, the acceleration is backward too, and the car slows down.
Using signs keeps this straight. With forward as positive, a backward net force is negative, and so is the acceleration. The law works the same way whether the object is speeding up or slowing down.
Isaac Newton published his three laws of motion in 1687, in a book usually called the Principia. The first law says an object keeps its velocity unless a net force acts. The second law, this lesson's, says how much a net force changes the velocity. The third says forces come in equal and opposite pairs.
The laws were so successful that they are still used to design bridges, cars, airplanes and rockets more than three hundred years later. Engineers at NASA plan spacecraft trajectories with the same law you use to find a cart's acceleration, just with bigger numbers and more careful arithmetic.
Mass works both ways. A heavy object is hard to get moving, and once it moves, it is hard to stop. A loaded freight train needs a huge net force to speed up, and the same huge force, from its brakes, to slow down.
That is why freight trains take over a kilometer to stop, and why railroad crossings have gates and warning lights well before a train arrives. No driver should ever try to beat a train to a crossing: the train simply cannot stop in time.
Even a tiny net force accelerates a huge mass, just slowly. A tugboat can push a giant cargo ship because, with the water's resistance small at low speed, the ship gradually speeds up.
At Chicago O'Hare or Atlanta's Hartsfield-Jackson airport, a Boeing 737 at the start of its takeoff roll has engines pushing with about two hundred forty kilonewtons, while its mass, loaded with passengers and fuel, is about seventy metric tons. Newton's second law gives an acceleration of a little over three meters per second squared.
That is about as brisk as a family car pulling away from a light. The plane keeps accelerating for half a minute or more, reaching about seventy meters per second, the speed needed to lift off. A fully loaded jumbo jet has more thrust but far more mass, so its acceleration is lower and it needs a longer runway, which is why the biggest planes use the longest runways.
Car safety engineers at the National Highway Traffic Safety Administration test seat belts and car seats on sleds that are rapidly accelerated to simulate a crash. Newton's second law tells them how large a force the equipment must survive.
If a thirty-kilogram child seat must withstand a deceleration of about two hundred meters per second squared, the belts and anchors must supply a force of six thousand newtons. Engineers use $F = ma$ to set the strength every buckle, strap and bolt must have, so that the seat holds the child in place when the car stops suddenly.
It is tempting to plug the largest force into $F = ma$, such as the push on a crate. But only the net force produces acceleration. With friction pushing back, part of the push is used up just canceling friction, and only the rest accelerates the crate.
A related error is to think a heavy object falls faster because gravity pulls it harder. It does pull harder, but the heavy object also has more mass to accelerate, and the two effects cancel exactly.
A $30$ N net force pushes a $6$ kg sled. Write the law.
$a = \dfrac{F}{m}$
Force over mass.
Substitute the values.
$a = \dfrac{30}{6}$
Newtons over kilograms.
Evaluate the acceleration.
$a = 5\ \text{m/s}^2$
Along the push.
Find it with a $12$ kg sled.
$a = \dfrac{30}{12} = 2.5\ \text{m/s}^2$
Twice the mass, half the acceleration.
Find it with a $60$ N net force on the $6$ kg sled.
$a = \dfrac{60}{6} = 10\ \text{m/s}^2$
Twice the force, twice the acceleration.
A student pushes a $25$ kg box with $120$ N against $70$ N of friction. Find the net force.
$F_{\text{net}} = 120 - 70 = 50\ \text{N}$
Opposite forces subtract.
Find the acceleration.
$a = \dfrac{50}{25} = 2\ \text{m/s}^2$
Net force over mass.
Find the speed after $3$ s from rest.
$v = 2 \times 3 = 6\ \text{m/s}$
$v = at$.
Find the acceleration if friction were ignored.
$a = \dfrac{120}{25} = 4.8\ \text{m/s}^2$
Too large.
Find the push for steady speed.
$P = 70\ \text{N}$
Net force zero.
Find the push for $3$ m/s².
$P = 70 + 25 \times 3 = 145\ \text{N}$
Friction plus $ma$.
A cart accelerates at $4$ m/s² under a $36$ N net force. Write the law for mass.
$m = \dfrac{F}{a}$
Rearranged.
Substitute the values.
$m = \dfrac{36}{4}$
Newtons over m/s².
Evaluate the mass.
$m = 9\ \text{kg}$
Kilograms.
A $3$ kg box is added. Find the new mass.
$m = 12\ \text{kg}$
Cart plus box.
Find the new acceleration with the same force.
$a = \dfrac{36}{12} = 3\ \text{m/s}^2$
Less than before.
Find the force to restore $4$ m/s².
$F = 12 \times 4 = 48\ \text{N}$
More mass needs more force.
Check the ratio.
$\dfrac{48}{36} = \dfrac{12}{9}$
Force grows with mass.
Write the law.
$F = ma$
Newton's second law.
Substitute the values.
$F = 15 \times 2$
Mass times acceleration.
Evaluate the force.
A net force of $150$ N acts on a $60$ kg cart. What is the cart's acceleration, in m/s²?
Complete the worked solution: a worker pushes a $15$ kg crate from rest with $70$ N across a floor where friction is $25$ N. Find the net force in N, the acceleration in m/s², and the speed after $4$ s in m/s.
Find the net force.
$F_{\text{net}} = P - f =$ n
Opposite forces subtract.
Find the acceleration.
$a = \dfrac{F_{\text{net}}}{m} =$ a
Newton's second law.
Find the speed after four seconds.
$v = a \times 4 =$ v
From rest.
Explain why the push is not used alone.
$\text{friction cancels part of it}$
Only the net force accelerates.
Match each statement to its meaning.
| net force equals mass times acceleration | half the acceleration | twice the acceleration | the force that gives 1 kg an acceleration of 1 m/s² | |
|---|---|---|---|---|
| Newton's second law | ||||
| the same force on twice the mass | ||||
| twice the net force on the same mass | ||||
| one newton |
A net force of $45$ N pushes a $15$ kg cart that starts at rest. Fill in its acceleration in m/s², its speed after $2$ s in m/s, and the net force in N needed to give it twice the acceleration.
| value | |
|---|---|
| acceleration (m/s²) | |
| speed after the time (m/s) | |
| force for double acceleration (N) |
A robot arm always pushes packages with the same net force, $24$ N. Write a package's acceleration, in m/s², as a function of its mass $m$ in kilograms.
Answer:
What net force, in N, gives a cyclist and bike of mass $80$ kg an acceleration of $1.5$ m/s²?
Answer: N
At the start of its takeoff roll, a Boeing 777 has a net forward force of about $1000$ kN from its engines and a mass of about $350$ metric tons. What is its acceleration, in m/s²?
Answer: m/s²
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A robot arm always pushes packages with the same net force, $150$ N. Write a package's acceleration, in m/s², as a function of its mass $m$ in kilograms.
Answer:
You can use Newton's second law. Explain to someone why a full shopping cart speeds up more slowly than an empty one pushed just as hard.
28. Your turn: what net force gives a $15$ kg wagon an acceleration of $2$ m/s²?, step 3
$F = 30\ \text{N}$
Net force.