Centres of Mass

The centre of mass is the point at which the weight of the object may be considered to act.

For a uniform object, the centre of mass is at the centre where lines of symmetry intersect.

For a uniform rod, the centre of mass is at its midpoint.

For a uniform rectangular lamina, the centre of mass is where the diagonals intersect.

For a non-uniform object, the centre of mass is not necessarily at the geometric centre.

For a uniform rod, draw the weight at the midpoint.

If a rod rests on supports, there are normal reaction forces perpendicular to the rod at each support.

If suspended by strings, there are tensions in the strings.

When there are two supports with unknown reactions, choose the pivot at one support to find the force at the other.

Step 1: Label the weight of the rod at a point (the centre of mass) on the rod.

Step 2: Call the perpendicular distance between the pivot and the weight x (or any letter).

Step 3: Form a moment equation using the fact that the rod is in equilibrium.

Step 4: Solve the equation to find x.

Read the question carefully to see which distance is asked for (e.g. from A or from B).

A rod tilts (rotates) about a pivot if the resultant moment about that pivot is non-zero.

When a rod is on the point of tilting about a pivot, the reaction forces at the other supports are zero.

The rod is still in equilibrium at the point of tilting.

Taking moments about the pivot allows you to ignore the reaction force at that pivot.

Examples: a person walking a plank, children on a seesaw.