A uniform rod of mass 0.5 kg and…

A uniform rod of mass 0.5 kg and length 0.6 m hangs vertically from a smooth horizontal pivot at its upper end. A horizontal light spring of…

One or more options correct

A uniform rod of mass

0.5\ \mathrm{kg}

and length

0.6\ \mathrm{m}

hangs vertically from a smooth horizontal pivot at its upper end. A horizontal light spring of stiffness

4\ \mathrm{N\,m^{-1}}

joins the lower end of the rod to a vertical wall and is unstretched when the rod is vertical. The rod is given small oscillations in the plane containing the spring. Take

g=10\ \mathrm{m\,s^{-2}}

. Which of the following statements is/are correct?

A. The angular frequency of small oscillations is $7\ \mathrm{rad\,s^{-1}}$ .
B. If the spring were removed, the period of small oscillations would be $2\pi/5\ \mathrm{s}$ .
C. For an angular amplitude of $0.1$ rad, the maximum speed of the lower end is $0.35\ \mathrm{m\,s^{-1}}$ .
D. The spring supplies more than half of the total restoring torque per unit angular displacement.

Show answer and solution

Answer: A, B

For a small angle

\theta

the restoring torque is

\tau=-\left(\dfrac{mgL}{2}+kL^2\right)\theta

(gravity acting at the centre, spring force

kL\theta

at the lower end). With

I=\dfrac{mL^2}{3}

\omega^2=\dfrac{3g}{2L}+\dfrac{3k}{m}=25+24=49

, so

\omega=7\ \mathrm{rad\,s^{-1}}

and (A) is correct. Without the spring

\omega^2=25

, so

T=2\pi/5\ \mathrm{s}

and (B) is correct. The maximum angular speed is

\theta_0\omega=0.7\ \mathrm{rad\,s^{-1}}

, so the lower end moves at

L\theta_0\omega=0.6\times0.7=0.42\ \mathrm{m\,s^{-1}}

, not

0.35

; (C) is wrong. The torque constants are

mgL/2=1.5\ \mathrm{N\,m}

for gravity and

kL^2=1.44\ \mathrm{N\,m}

for the spring, so the spring gives only about

49\%

; (D) is wrong. Correct options: A and B.

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Original question written in the JEE Advanced 2017 Paper 1 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake