The GATE Grind

GATE 2024 CE (CE2) – Question 16

Engineering Mechanics · System of forces, free-body diagrams, equilibrium equations · 1 mark · Multiple choice

A 3 m long, horizontal, rigid, uniform beam PQ has negligible mass. The beam is subjected to a 3 kN concentrated vertically downward force at 1 m from P, as shown in the figure. The beam is resting on vertical linear springs at the ends P and Q. For the spring at the end P, the spring constant $K_P = 100$ kN/m.

[Figure: the beam PQ of length 3 m with the 3 kN load at 1 m from P (and 2 m from Q) and a spring $K_P$ at P and $K_Q$ at Q.]

If the beam DOES NOT rotate under the application of the force and displaces only vertically, the value of the spring constant $K_Q$ (in kN/m) for the spring at the end Q is

Diagram for GATE 2024 CE (CE2) question 16
  1. 150
  2. 100
  3. 50
  4. 200

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (C) 50

Explanation

Take moments about Q: $R_P \times 3 = 3 \times 2$, so $R_P = 2$ kN and $R_Q = 1$ kN. The beam does not rotate, so both ends move down by the same amount: $\delta = \frac{R_P}{K_P} = \frac{2}{100} = 0.02$ m. Then $K_Q = \frac{R_Q}{\delta} = \frac{1}{0.02} = 50$ kN/m.