Newton’s laws and changes in motion

When forces balance, the resultant force is zero and there is no acceleration. A book resting on a table has a downward gravitational force balanced by an upward contact force from the table. A car travelling at constant velocity has a forward driving force balanced by friction and air resistance. Both objects have forces acting on them, but neither changes its velocity.

A moving object does not need a resultant force to keep moving. A bicycle slows when the rider stops pedalling because friction and air resistance produce a backward resultant force. Without that resultant force, it would continue at constant velocity.

A non-zero resultant force causes acceleration in the direction of the resultant force. A forward resultant can make a moving car speed up; a backward resultant can make it slow down. A force acting across the direction of travel can change the direction of motion.

Free fall and the size of everyday accelerations

In free fall, gravity is the only force acting: air resistance is absent or negligible. Near Earth's surface, use the acceleration due to gravity:

$$g=10\text{ m/s}^2$$

This means a falling object's downward velocity increases by about $10\text{ m/s}$ every second. Without air resistance, objects of different masses have the same acceleration.

To estimate an everyday acceleration, estimate the change in velocity and divide it by the time taken. A typical family car might increase its speed from rest to about $27\text{ m/s}$ in $10\text{ s}$, giving an average acceleration of $27/10=2.7\text{ m/s}^2$: a few metres per second squared, smaller than free-fall acceleration.

For a simple walking estimate, suppose someone starts from rest and reaches $1.5\text{ m/s}$ in $3\text{ s}$. Their average acceleration would be $0.5\text{ m/s}^2$. These estimates are not fixed values for every person or vehicle; they help you judge a sensible magnitude.

Newton's second law: how much does motion change?

Newton's second law connects resultant force, mass and acceleration:

$$F=ma$$

Here, $F$ is resultant force in newtons, $m$ is mass in kilograms and $a$ is acceleration in metres per second squared. One newton gives a mass of $1\text{ kg}$ an acceleration of $1\text{ m/s}^2$.

For a fixed mass, doubling the resultant force doubles the acceleration: acceleration is directly proportional to resultant force. For a fixed resultant force, doubling the mass halves the acceleration: acceleration is inversely proportional to mass.

The equation can be rearranged to find either acceleration or mass:

$$a=\frac{F}{m}\qquad m=\frac{F}{a}$$

For example, a $1500\text{ kg}$ van has a driving force of $4200\text{ N}$ and resistance of $1200\text{ N}$. First find the resultant: $4200-1200=3000\text{ N}$ forwards. Then $a=3000/1500=2.0\text{ m/s}^2$ forwards. The driving force alone would give the wrong answer because resistance also acts.

For example, a $2.0\text{ kg}$ trolley accelerating at $0.50\text{ m/s}^2$ requires a resultant force of $F=2.0\times0.50=1.0\text{ N}$. If an object instead accelerates at $0.50\text{ m/s}^2$ under a $3.0\text{ N}$ resultant force, its mass is $m=3.0/0.50=6.0\text{ kg}$.

Core practical: investigating force, mass and acceleration

A trolley attached by a light string to hanging masses can be used to investigate Newton's second law. The string passes over a pulley, so the falling masses pull the trolley along a runway. A motion sensor and data logger can record a velocity–time graph; its gradient gives acceleration.

Measure the masses using a balance. The mass being accelerated includes the trolley, any added masses and the hanger with its masses: they move together as one system. Secure the runway and pulley, keep the string taut and parallel to the runway, and use a stop block to prevent the trolley hitting the pulley. Keep feet clear of the falling hanger.

To investigate mass, keep the hanging mass unchanged and add known masses to the trolley. The hanging weight then provides approximately the same pulling force on the whole system, while its total mass increases.

Release the trolley from the same marked position without pushing it. Record its acceleration, repeat each measurement and calculate a mean. Repeat for several different added masses, recording the total moving mass each time. Keep the runway angle, pulley arrangement and release position unchanged.

At approximately constant resultant force, increasing the total moving mass decreases acceleration. An acceleration–mass graph is a downward curve; plotting acceleration against $1/m$ gives a straight line through the origin for the ideal relationship.

To investigate force, transfer masses from the trolley to the hanger rather than introducing extra masses. This increases the hanging weight while keeping the total moving mass constant. Repeat the acceleration measurements for each force. Ideally, an acceleration–resultant-force graph is a straight line through the origin.

Friction and pulley resistance oppose motion, so the hanging weight is not exactly the resultant force on the whole system. A low-friction runway helps the results approach the ideal relationships. Releasing consistently and repeating measurements improves the reliability of the comparison.

Higher tier: constant speed in a circular orbit

An object moving in a circular orbit at constant speed still has a continuously changing velocity because its direction continually changes. At each point, its velocity is along the tangent to the circle—the direction it is travelling at that instant.

Changing velocity means the object is accelerating. Circular motion therefore needs a non-zero resultant force towards the centre of the circle, called the centripetal force. This inward force continually turns the direction of motion rather than increasing the speed. Gravity provides the centripetal force for a satellite in orbit.

Centripetal force describes the role of the resultant force, not a new type of force. If the inward force disappeared, the object would travel along a straight-line tangent rather than continuing around the circle.

Higher tier: inertial mass

Inertial mass measures how difficult it is to change an object's velocity, including getting it moving from rest. It is defined as the ratio of resultant force to acceleration:

$$m=\frac{F}{a}$$

A larger inertial mass means a smaller acceleration for the same resultant force. An empty trolley is easier to accelerate than a heavily loaded trolley: the loaded trolley has greater resistance to a change in velocity.

• First law: zero resultant force → rest or constant velocity.
• Non-zero resultant force → changing speed and/or direction; acceleration is in the resultant force's direction.
• Free fall near Earth: $g=10\text{ m/s}^2$, with air resistance absent or negligible.
• Estimate acceleration using change in velocity ÷ time. A family car accelerating normally has an acceleration of a few $\text{m/s}^2$.

Newton's second law

$$F=ma\qquad a=\frac{F}{m}\qquad m=\frac{F}{a}$$

$F$: resultant force in N; $m$: mass in kg; $a$: acceleration in $\text{m/s}^2$.

• Fixed mass: $a\propto F$.
• Fixed resultant force: $a\propto 1/m$.

Core practical

• Trolley, string, pulley and hanging masses; measure acceleration using velocity–time data.
• Change mass: add masses to trolley; keep hanging mass fixed.
• Change force: transfer masses from trolley to hanger; keep total moving mass fixed.
• Release without pushing, keep the arrangement unchanged, repeat and calculate means.
• Include all moving masses; friction reduces the resultant force.

Higher tier

• Circular motion at constant speed has changing velocity because direction changes.
• A resultant centripetal force acts towards the centre.
• Inertial mass measures resistance to changes in velocity: $m=F/a$.