1.17-1.18 - Newton's second law and weight

Force, mass and acceleration

Newton's second law relates the resultant force to acceleration. If the resultant force is zero, acceleration is zero: an object may remain at rest or continue at constant velocity. In this equation, the force is the resultant force along the line of motion, not every force added without direction.

For the same mass, a larger resultant force produces a larger acceleration. If the same resultant force acts on a larger mass, the acceleration is smaller: the same push speeds up an empty shopping trolley more quickly than a full one.

The Edexcel relationship is:

where:

• $F$ is the unbalanced force in newtons, $N$
• $m$ is the mass in kilograms, $kg$
• $a$ is the acceleration in metres per second squared, $m/s^2$

This equation also explains the size of the newton: $1 N$ is the force needed to give a $1 kg$ mass an acceleration of $1 m/s^2$.

Using F = m x a

Use the resultant force, not an individual push or pull, to calculate acceleration. Convert a mass in grams to kilograms before substituting: for example, 250 g is 0.250 kg, so a resultant force of 1.0 N gives an acceleration of 1.0/0.250 = 4.0 m/s².

For Newton's second law:

• to find force: $F = m \times a$
• to find mass: $m = F / a$
• to find acceleration: $a = F / m$

Worked example:

A cyclist and bicycle have a total mass of $80 kg$. The resultant forward force is $120 N$. Calculate the acceleration.

Use:

Substitute:

The acceleration is $1.5 m/s^2$ in the direction of the resultant force.

Be careful with the wording. If a question gives two forces acting in opposite directions, find the resultant first. For example, a forward driving force of $50 N$ and a backward friction force of $20 N$ give an unbalanced force of $30 N$ forwards. Use this $30 N$ as $F$ in $F = m \times a$, not the $50 N$ driving force, and never the sum $70 N$.

Weight, mass and gravitational field strength

Mass and weight are connected, but they are not the same quantity.

Mass is measured in kilograms. It tells you how much matter an object has, and it is the same for the object unless matter is added or removed.

Weight is a force. It is the gravitational force on a mass, so it is measured in newtons. Because it is a force, weight is a vector: it acts downwards, towards the centre of the Earth. Mass is a scalar: it has size but no direction.

In everyday speech people say "I weigh 52 kg". In physics, $52 kg$ is a mass. Weight is a force, so it must be given in newtons.

Gravitational field strength, $g$, tells you the weight per kilogram of mass at a particular place. Its unit is newtons per kilogram, $N/kg$.

where:

• $W$ is weight in newtons, $N$
• $m$ is mass in kilograms, $kg$
• $g$ is gravitational field strength in newtons per kilogram, $N/kg$

Near the surface of Earth, $g$ is about $9.8 N/kg$, and questions often round it to $10 N/kg$; use the value given in the question. On a planet or moon with a smaller gravitational field strength, the same mass has a smaller weight. For example, a $60 kg$ astronaut weighs $60 \times 10 = 600 N$ on Earth but only $60 \times 1.6 = 96 N$ on the Moon, where $g$ is about $1.6 N/kg$. Their mass is still $60 kg$.

Using W = m x g

The weight equation works in the same way as other formulae: write it down, substitute, then calculate with the correct unit.

Worked example:

A student has mass $52 kg$. The gravitational field strength is $9.8 N/kg$. Calculate the student's weight.

Use:

Substitute:

The student's weight is about $510 N$.

You may also need to rearrange:

• to find mass: $m = W / g$
• to find gravitational field strength: $g = W / m$

Worked example:

An object has weight $180 N$ where the gravitational field strength is $6.0 N/kg$. Its mass is:

The mass is $30 kg$. Do not write $30 N$; newtons are for forces such as weight.

Choosing the right equation

The two equations have a similar shape, but they answer different questions.

Use $F = m \times a$ when the question is about an unbalanced force causing acceleration. The acceleration is the actual change in velocity per second.

Use $W = m \times g$ when the question is about the force of gravity on a mass. The gravitational field strength describes force per kilogram; it is not necessarily the object's actual acceleration.

The symbols also carry meaning:

• $F$ can be any unbalanced force on an object
• $W$ is specifically the object's weight
• $a$ is acceleration caused by an unbalanced force
• $g$ is gravitational field strength at a place

If weight is the only force, the two equations connect: ma = mg, so a = g. For example, a $2.0 kg$ ball falling where $g = 10 N/kg$, with no air resistance, has weight $2.0 \times 10 = 20 N$, so its acceleration is $a = F / m = 20 / 2.0 = 10 m/s^2$. This also explains why N/kg and m/s² are equivalent units. A supported object still has weight even when it has zero acceleration: its upward support balances its downward weight.

Quick recap

• Newton's second law is $F=ma$, using the resultant force. At fixed mass, more resultant force gives more acceleration; at fixed force, more mass gives less acceleration.
• Zero resultant force means zero acceleration, so an object can remain at rest or continue at constant velocity.
• Weight is $W=mg$. Mass is measured in kg, weight in N and gravitational field strength in $\mathrm{N/kg}$.
• Use $F=ma$ for acceleration and $W=mg$ for gravitational force. An object can have weight while its forces balance.