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 $$\underline{1.5m}$$.