← 2020 Paper 1

UPSC 2020 Maths Optional Paper 1 Q6c — Step-by-Step Solution

15 marks · Section B

Equilibrium of a system of particles · Dynamics & Statics · asked 8× in 14 yrs · Read the full method →

Question

A beam $AD$ rests on two supports $B$ and $C$, where $AB=BC=CD$. It is found that the beam will tilt when a weight of $p$ kg is hung from $A$ or when a weight of $q$ kg is hung from $D$. Find the weight of the beam.

Technique

Limiting equilibrium / tilting: at the verge of tilting the far support reaction is zero; take moments about the near support. Two cases, then eliminate the unknown CG position.

Solution

Setup

Let $AB=BC=CD=\ell$, so measuring from $A$:

Let the beam’s weight be $W$, acting at its centre of gravity $G$ at distance $x$ from $A$. (We do not assume the beam is uniform; if the result forces $x=\tfrac{3\ell}{2}$ it would mean $p=q$, but generally $p\ne q$, so $x$ is unknown too.)

The beam rests on the two supports $B$ and $C$, which can only push up. When a load is hung, one support’s reaction can drop to zero; the beam is on the verge of tilting about the other support.

Step 1 — Tilt when $p$ is hung from $A$

Hanging $p$ at $A$ tends to lift the $D$–end, so the beam is about to rotate (tilt) about $B$; the reaction at $C$ just vanishes, $R_C=0$.

Take moments about $B$ (the pivot). The downward load $p$ at $A$ is at horizontal distance $\ell$ on the $A$–side; the beam weight $W$ at $G$ is at distance $(x-\ell)$ on the $D$–side. At the verge of tilting these balance:

Step 2 — Tilt when $q$ is hung from $D$

Hanging $q$ at $D$ tends to lift the $A$–end, so the beam is about to tilt about $C$; $R_B=0$.

Moments about $C$: load $q$ at $D$ is at distance $\ell$; weight $W$ at $G$ is at distance $(2\ell-x)$ on the $A$–side:

Step 3 — Eliminate $x$

Add (1) and (2):

Therefore

Answer