Tilting

In a nutshell

Some bodies will be on the point of tilting about a pivot, and using your knowledge of reaction forces you can work out unknowns in a problem.

Tilting

When a body is on the point of tilting about a pivot, the reaction force at any other support, or tension in a wire, is zero. This allows you to eliminate one unknown, thus making it easier to calcuate any other unknowns.

Example 1

A uniform beam $AB$ with weight of $50N$ and a length of $20m$ rests on supports $C$ and $D$ . $AC=6m$ and $AD= 16m$ . A box is placed down on point $A$ and causes the beam to be on the point of tilting about $C$ . Find the weight of the box.

The beam is on the point of tilting, therefore the reaction force at $D$ is $0N$ . Taking moments about $C$ means $R_C$ can be ignored. Take the weight of the box to be $xN$ . The beam is uniform therefore the centre of mass is at the midpoint of $AB$ .

Equate clockwise and anticlockwise moments:

$\begin{aligned}x\times 6&= 50\times 4\\ 6x&=200\\ x&=\dfrac{200}{6} = 33.3\end{aligned}$ 

The box has a weight of $\underline{33.3N}$ .

Example 2

A non-uniform rod weighing $30N$ with a length of $15m$ rests on supports $A$ and $B$ . When a weight of $15N$ is added on the end of the rod $3m$ to the right of $B$ , the rod is on the point of rotating about $B$ . Find the distance of the centre of mass of the rod from point $B$ .

Assume the centre of mass gives an anticlockwise moment about point $B$ . Taking moments about $B$ , the reaction force at this point can be ignored. The reaction at point $A$ is $0$ .

Equate clockwise and anticlockwise moments:

$\begin{aligned}30x&=15\times 3\\30x&=45\\ x&=\dfrac{45}{30} =1.5\end{aligned}$ 

Therefore, the distance of the centre of mass of the rod from point $B$ is $\underline{1.5m}$ .