Newton’s second law: F = ma

Force, mass and acceleration

Newton's first law says what happens with no net force. The second law says what a net force does: it makes the object accelerate, in the direction of the net force.

• The bigger the net force, the bigger the acceleration (for the same mass). Double the force, double the acceleration.
• The bigger the mass, the smaller the acceleration (for the same force). Double the mass, halve the acceleration.

Newton's second law: net force = mass × acceleration
F = m a    so    a = F ÷ m    and    m = F ÷ a

With mass in kg and acceleration in m/s², force is in newtons. In fact, this is how the newton is defined: 1 N is the net force that gives a 1 kg mass an acceleration of 1 m/s². Remember that F is the net force: you must add up all the forces first.

Mass and weight

Mass is the amount of matter in an object, measured in kilograms. It is the same everywhere. Weight is the force of gravity on an object, measured in newtons. It comes straight from F = ma, using the acceleration due to gravity:

weight = mass × g    W = m g    (g = 9.8 N/kg on Earth)

A 100 g apple has a weight of 0.1 × 9.8 = 0.98 N, about 1 N. On the Moon, where g is 1.6 N/kg, the same apple weighs only 0.16 N. Gravity never completely disappears, even in deep space, though it gets weaker with distance. Astronauts on the International Space Station float not because there is no gravity there (it is about 90% as strong as on the ground), but because they and the station are falling around the Earth together.

Investigating F = ma

A trolley on a smooth bench can be pulled by a string over a pulley to a hanging mass. The weight of the hanging mass is the pulling force. Because the hanging mass accelerates too, the total mass being accelerated is the trolley plus the hanging mass. A motion sensor, light gate or ticker timer measures the acceleration.

Falling with air resistance

In Year 7 you saw that a skydiver reaches a steady terminal velocity. Newton's second law explains why. Just after jumping, the only force is weight, so a = g. As the skydiver speeds up, air resistance grows, so the net force (weight − air resistance) shrinks and so does the acceleration. At about 200 km/h in a belly-down position, air resistance equals weight: the net force is zero and the velocity stays constant. Opening the parachute suddenly makes air resistance much bigger than weight, so the net force is upwards and the skydiver decelerates to a new, much slower terminal velocity of about 20 km/h.