← 2020 Paper 1

UPSC 2020 Maths Optional Paper 1 Q7c-i — Step-by-Step Solution

10 marks · Section B

Principle of virtual work · Dynamics & Statics · asked 6× in 13 yrs · Read the full method →

Question

A square framework formed of uniform heavy rods of equal weight $W$ jointed together, is hung up by one corner. A weight $W$ is suspended from each of the three lower corners, and the shape of the square is preserved by a light rod along the horizontal diagonal. Find the thrust of the light rod.

Technique

Principle of virtual work with the single DOF $\theta$ (half-angle of the rods); $T=(dV/d\theta)/(dL/d\theta)$.

Solution

Setup

Label the square $ABCD$ hung from the top corner $A$, so the diagonal $AC$ is vertical and the diagonal $BD$ is horizontal. The four uniform rods $AB, BC, CD, DA$ each have weight $W$. Each rod makes $45^\circ$ with the vertical (square geometry). Hung weights $W$ act at the three lower corners $B, C, D$. A light rod lies along the horizontal diagonal $BD$.

Use the principle of virtual work. Let each upper rod make angle $\theta$ with the downward vertical (the square configuration is $\theta=45^\circ$), and let each rod have length $2a$. Take $A$ as origin, $y$ measured upward (so the frame is at $y<0$).

Step 1 — Heights of all weights as functions of $\theta$

Going down the rods at angle $\theta$ to the vertical, each rod’s vertical extent is $2a\cos\theta$.

• Midpoints (CGs) of the two upper rods $AB,AD$: depth $a\cos\theta$, i.e. $y=-a\cos\theta$ each.
• Corners $B,D$: depth $2a\cos\theta$, $y=-2a\cos\theta$.
• Midpoints of the two lower rods $BC,DC$: depth $3a\cos\theta$, $y=-3a\cos\theta$ each.
• Bottom corner $C$: depth $4a\cos\theta$, $y=-4a\cos\theta$.

Step 2 — Total potential energy

Weights: four rod weights $W$ at the rod midpoints, plus hung weights $W$ at $B$, $C$, $D$.

Step 3 — The diagonal $BD$ and virtual work

Length of the horizontal diagonal:

Let the thrust (compressive force) in the light rod be $T$; it pushes the corners $B$ and $D$ apart, so it does positive work when $L_{BD}$ increases. The principle of virtual work for the single degree of freedom $\theta$:

Step 4 — Evaluate at the square configuration $\theta=45^\circ$

Answer