Mass, weight and centre of mass

​​In a nutshell

Mass is a measure of how much matter there is in an object, while weight is the gravitational force that acts on an object as a result of its mass. The centre of mass of an object is the average position of the mass of an object, while the centre of gravity is a point that the weight of an object acts through. They are the same in a uniform gravitational field.

Equations

Constants

Variable definitions

Definitions

Mass and weight

​​Mass and force

Mass is a property of an object which quantifies how much matter there is in the object. Force, acceleration and mass are related by the following equation.

​$$F = m a$$​​

Where $$F$$ is the force, $$m$$​ is the mass and $$a$$ is the acceleration.​

The more mass an object has, the stronger the force required to give an object a certain acceleration.

Example

An object experiences a force of $$10N$$​ and accelerates at $$5ms^{-2}$$ as a result. Calculate the object's mass.

First, write down the equation.

$$F = m a$$​​

Next, rearrange the equation to find mass.

$$m = \dfrac{F}{a}$$​​

Finally, substitute in the values.

$$m = \dfrac{10}{5}$$​​

$$m = 2\,kg$$​​

The mass of the object is $$\underline{2\,kg}$$.

Weight

The weight on an object is the gravitational force an object experiences as a direct result of its mass. Using the relationship between force, mass and acceleration, it is possible to derive an equation for the weight of an object. In this case, the force is the weight of an object, the mass is the object's mass and the acceleration is the gravitational acceleration.

These quantities can be substituted into the relationship, producing the following equation.

​$$W = mg$$​​

Where $$W$$​ is the weight and $$g$$ is the gravitational acceleration. On Earth, $$g$$ has a value of ​$$9.81\,ms^{-2}$$.

Curiosity: While an object's mass doesn't change, its weight can technically range from zero to infinite, depending on the strength of the gravitational field it is located in. The weight of an object on the moon is $$6$$ times lower than if it were on Earth!

Example

Calculate the weight of an object with a mass of $$2\,kg$$.

First, write down the equation.

$$W = mg$$​​

Next, substitute in the values.

$$W = 2 \times 9.81$$​​

$$W = 19.62\,N$$​​

The weight of the object is $$\underline{19.62}N$$.

Remember: Mass is how much matter is in an object and is measured in kilograms $$kg$$, while weight is the gravitational force on an object measured in Newtons $$N$$.

Centre of mass and centre of gravity

The centre of mass of an object is a point that can be considered as an average position of the mass in an object. The centre of gravity usually coincides with the centre of mass, and is a point where the resultant gravitational force acts through.

​​Example

This is a diagram of an irregular object. The blue arrows are the weight forces acting on different parts of the object, while the red arrow is the resultant weight force. The red dot is the centre of mass of the object.