Question

The resultant of two like parallel forces $P$ and $Q$ passes through a point $O$. If the resultant also passes through $O$ when $Q$ and $R$ replace $P$ and $Q$ respectively, then

• A. $P,Q,R$ are in GP
• B. $Q,P,R$ are in GP
• C. $R,P,Q$ are in GP
• D. $P,Q,R$ are in AP

Answer 1

The correct option is A $P,Q,R$ are in GP

Let the two like parallel forces $P$ and $Q$ act at points $A$ and $B$ on the rigid body. Since, the resultant of $P$ and $Q$ passes through $O$.
Therefore,
$AO=\left[\frac{AB}{P+Q}\right]Q$ ...(i)
The resultant of like parallel forces $Q$ and $R$, acting at $A$ and $B$ respectively, also passes through $O$. Therefore,
$AO=\left(\frac{AB}{Q+R}\right)R$ ....(ii)
From equations (i) and (ii), we get
$\left(\frac{AB}{P+Q}\right)Q=\left(\frac{AB}{Q+R}\right)R$
$\Rightarrow Q^2+QR=PR+QR$
$\Rightarrow Q^2=PR$
$\Rightarrow P,Q$ and $R$ are in GP.