3.2 Forces, acceleration and Newton's laws of motion — Learn

Prerequisites: Section 3.1, especially velocity and acceleration; Section 2.1, especially forces, resultant forces, mass and weight; substituting into a formula and rearranging it.

By the end of this section you can: state and use Newton's three laws of motion; calculate a resultant force, a mass or an acceleration using $F = ma$ ; and, at Higher tier, explain what inertial mass means.

1. Newton's first law

A resultant force is the single force that has the same effect as all the forces acting on an object together. If the resultant force on an object is zero, the forces are balanced.

Newton's first law says:

if the resultant force on a stationary object is zero, the object stays stationary;
if the resultant force on a moving object is zero, the object continues to move at the same speed and in the same direction.

So an object keeps a steady velocity until a resultant force acts on it. A car that travels at a steady speed along a straight road has a driving force from the engine, and resistive forces from friction and air resistance. The two are equal and opposite, so the resultant force is zero.

2. Newton's second law

A resultant force changes the velocity of an object. The object accelerates.

Newton's second law says that the acceleration of an object is proportional to the resultant force, and inversely proportional to the mass of the object.

a \propto F \qquad a \propto \frac{1}{m}

The symbol $\propto$ means "is proportional to". As an equation,

F = m a

where $F$ is the resultant force in newtons, $m$ is the mass in kilograms, and $a$ is the acceleration in metres per second squared.

Worked example 1. A car of mass 1200 kg accelerates at $2.5\,\text{m/s}^2$ . Calculate the resultant force.

\begin{aligned} F &= m a \\ &= 1200 \times 2.5 \\ &= 3000\,\text{N} \\ &= 3.0 \times 10^3\,\text{N} \end{aligned}

Worked example 2. A resultant force of 60 N acts on an object. The acceleration is $4.0\,\text{m/s}^2$ . Calculate the mass of the object.

\begin{aligned} m &= \frac{F}{a} \\ &= \frac{60}{4.0} \\ &= 15\,\text{kg} \end{aligned}

Worked example 3. A resultant force of 240 N acts on a trolley of mass 80 kg. Calculate the acceleration.

\begin{aligned} a &= \frac{F}{m} \\ &= \frac{240}{80} \\ &= 3.0\,\text{m/s}^2 \end{aligned}

Watch out. Use the resultant force in $F = ma$ , not one of the individual forces. If a car is pushed forwards with 800 N and air resistance is 300 N backwards, the resultant force is $800 - 300 = 500\,\text{N}$ . Using 800 N gives an acceleration that is too large. Also check that the mass is in kilograms: 500 g is 0.5 kg, not 500 kg. Re-check by asking whether your answer is sensible for the object in front of you.

Try it: force, mass and acceleration

The figure shows a trolley with a resultant force acting on it. You can change the force and the mass, and read the acceleration.

Predict first. Do not move the sliders yet. Write down your answers.

If you double the resultant force, what happens to the acceleration?
If you double the mass, what happens to the acceleration?

Test. Set the force to 8 N and the mass to 4 kg. Read the acceleration. Now double the mass and read the acceleration again.

Explain. For the same force, why does doubling the mass halve the acceleration? Write your answer in two or three sentences. Use the equation $F = ma$ .

3. Inertia and inertial mass

The tendency of an object to stay at rest, or to keep moving at a constant velocity, is called inertia. A heavy object is harder to start moving and harder to stop than a light one.

Inertial mass is a measure of how difficult it is to change the velocity of an object. It is defined as the ratio of the force acting on the object to the acceleration it produces:

\text{inertial mass} = \frac{\text{force}}{\text{acceleration}} \qquad m = \frac{F}{a}

Worked example 4. A resultant force of 250 N gives an object an acceleration of $0.50\,\text{m/s}^2$ . Calculate the inertial mass of the object.

\begin{aligned} m &= \frac{F}{a} \\ &= \frac{250}{0.50} \\ &= 500\,\text{kg} \end{aligned}

4. Newton's third law

Newton's third law says that whenever two objects interact, the forces they exert on each other are equal and opposite.

The two forces act on different objects, so they do not cancel. If you push a wall, your hand pushes the wall and the wall pushes your hand back with the same size force in the opposite direction.

Here are two examples of equilibrium. A book rests on a table: the weight of the book pulls down on the table, and the table pushes up on the book with an equal force. A swimmer pushes the water backwards, and the water pushes the swimmer forwards.

5. Estimating large accelerations and forces

You can use $F = ma$ to estimate the force in an everyday situation. This helps you judge whether a force is safe.

Worked example 5. A car of mass 1500 kg accelerates at $6.0\,\text{m/s}^2$ . Calculate the resultant force on the car.

\begin{aligned} F &= m a \\ &= 1500 \times 6.0 \\ &= 9000\,\text{N} \\ &= 9.0 \times 10^3\,\text{N} \end{aligned}

A force of 9000 N is large. Large forces and large accelerations are why safety features matter. Section 3.3 looks at how to reduce the danger in a crash.

6. The force and acceleration investigation

You can investigate the effect of the force on the acceleration of a trolley of constant mass.

Set up a trolley on a runway. Attach a string that passes over a pulley to a hanging mass. The hanging weight is the force that pulls the trolley.
Use light gates or a motion sensor to measure the velocity of the trolley as it passes, and find the acceleration from a velocity-time graph.
Keep the mass of the trolley constant. Increase the hanging mass to increase the force, and record the acceleration each time.
Then keep the force constant and add masses to the trolley. Record the acceleration each time.

A graph of acceleration against force is a straight line through the origin. A graph of acceleration against mass is a curve that falls, and a graph of acceleration against $\dfrac{1}{\text{mass}}$ is a straight line through the origin. This is the evidence behind Newton's second law.

Progress check

Return to the chapter problem. The bus and its passengers have a mass of 12000 kg. It accelerates from rest to 12 m/s in 8.0 s. Work out the acceleration, then work out the resultant force on the bus.

You should now be able to

use Newton's first law to explain the motion of an object with a zero resultant force;
use Newton's second law, $F = ma$ , to calculate force, mass or acceleration;
explain that acceleration is proportional to force and inversely proportional to mass;
define inertial mass as the ratio of force to acceleration (Higher tier);
use Newton's third law for examples of two objects interacting;
describe an investigation into force and acceleration;
estimate the force in a large acceleration.