What is the centre of mass?

The point at which the total mass can be taken to act.

What do you divide by?

The total mass, not the number of particles.

Formulae for the centre of mass in 2D?

\bar{x}=\frac{\sum mx}{\sum m} and \bar{y}=\frac{\sum my}{\sum m}.

Vector form of the centre of mass?

\bar{\mathbf{r}}=\dfrac{\sum mi\mathbf{r}i}{\sum mi}

Where does a light rod with particles balance?

At the centre of mass of the particles.

What is a good choice of origin?

A point that makes some terms zero, such as the position of one particle or one end of a rod.

What happens when a particle is added?

Add its moment to the sum and its mass to the total, then divide again.

How do you find an unknown mass or position?

Write \bar{x}=\frac{\sum mx}{\sum m} with the unknown, set it equal to the given value and solve.

How do you find the distance of the centre of mass from a point?

Use Pythagoras on the differences in x and y.

Centre of mass of discrete massesEdexcel International A Level Further Maths: Flashcards

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Formula for the centre of mass of particles on a line?

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Formula for the centre of mass of particles on a line?: $\bar{x}=\dfrac{\sum m_ix_i}{\sum m_i}$
What is the centre of mass?: The point at which the total mass can be taken to act.
What do you divide by?: The total mass, not the number of particles.
Formulae for the centre of mass in 2D?: $\bar{x}=\frac{\sum mx}{\sum m}$ and $\bar{y}=\frac{\sum my}{\sum m}$ .
Vector form of the centre of mass?: $\bar{\mathbf{r}}=\dfrac{\sum m_i\mathbf{r}_i}{\sum m_i}$
Where does a light rod with particles balance?: At the centre of mass of the particles.
Does a light rod contribute to the centre of mass?: No; it has no mass.
What is a good choice of origin?: A point that makes some terms zero, such as the position of one particle or one end of a rod.
Where is the centre of mass relative to two unequal masses?: Closer to the heavier mass.
What happens when a particle is added?: Add its moment to the sum and its mass to the total, then divide again.
How do you find an unknown mass or position?: Write $\bar{x}=\frac{\sum mx}{\sum m}$ with the unknown, set it equal to the given value and solve.
How do you find the distance of the centre of mass from a point?: Use Pythagoras on the differences in x and y.
What is the total moment about the centre of mass?: Zero.

Exam questions on Centre of mass of discrete masses

Particles of mass $2$ kg, $3$ kg and $5$ kg are placed on a straight horizontal line at distances $1$ m, $4$ m and $7$ m respectively from a fixed point $O$ , all on the same side of $O$ .
Instead, a fourth particle of mass $10$ kg is placed at a distance $d$ m from $O$ so that the centre of mass of the four particles is $6$ m from $O$ . Find $d$ .2 marks
Particles of mass $2$ kg, $3$ kg and $5$ kg are placed at the points $(1,4)$ , $(4,0)$ and $(3,2)$ respectively, where the coordinates are in metres relative to an origin $O$ .
Find the distance of the centre of mass from the origin $O$ .2 marks
A light rod $AB$ has length $4$ m. Particles of mass $3$ kg, $m$ kg and $5$ kg are attached to the rod at $A$ , at the midpoint of $AB$ and at $B$ respectively. The centre of mass of these three particles is $2.4$ m from $A$ .
Find the value of $m$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).