Angular SHM

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LEVELJEE Advanced

Two masses $m$ and $m /2$ are connected at the two ends of a massless rigid rod of length $l$ . The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system (see figure). Because of torsional constant $k$ , the restoring torque is $τ = k θ$ for angular displacement $θ$ . If the rod is rotated by $θ_{0}$ and released, the tension in it when it passes through its mean position will be

\frac{2 k θ _{0}^{2}}{l}

\frac{k θ _{0}^{2}}{l}

\frac{3 k θ _{0}^{2}}{l}

\frac{k θ _{0}^{2}}{2 l}

LEVELJEE Advanced

Two light identical springs of spring constant $k$ are attached horizontally at the two ends of an uniform horizontal rod AB of length $l$ and mass $m$ . The rod is pivoted at its centre 'O' and can rotate freely in horizontal plane. The other ends of the two springs are fixed to rigid supports as shown in figure. The rod is gently pushed through a small angle and released. The frequency of resulting oscillation is

\frac{1}{2 π} \frac{2 k}{m}

\frac{1}{2 π} \frac{3 k}{m}

\frac{1}{2 π} \frac{6 k}{m}

\frac{1}{2 π} \frac{k}{m}

JEE Advanced (1989)

LEVELJEE Advanced

A point particle of mass $M$ attached to one end of a massless rigid non-conducting rod of length $L$ . Another point particle of the same mass is attached to the other end of the rod. The two particles carry charges $+ q$ and $- q$ respectively. This arrangement is held in a region of a uniform electric field $E$ such that the rod makes a small angle $θ$ (say of about 5 degrees) with the field direction. Find an expression for the minimum time needed for the rod to become parallel to the field after it is set free.

JEE Advanced 2011

LEVELJEE Advanced

A metal rod of length $L$ and mass $m$ is pivoted at one end. A thin disc of mass $M$ and radius $R$ ( $R < L$ ) is attached at its centre to the free end of the rod. Consider two ways the disc is attached. Case A—the disc is not free to rotate about its centre and Case B—the disc is free to rotate about its centre. The rod-disc system performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is/are true?

* Multiple Correct Options

Restoring torque in case A = Restoring torque in case B

Restoring torque in case A < Restoring torque in case B

Angular frequency for case A > Angular frequency for case B

Angular frequency for case A < Angular frequency for case B

JEE Advanced 2023

LEVELJEE Advanced

Two point-like objects of masses 20 gm and 30 gm are fixed at the two ends of a rigid massless rod of length 10 cm. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $1.2 \times 1 0^{- 8} N m rad^{- 1}$ . The angular frequency of the oscillations is $n \times 1 0^{- 3} rad s^{- 1}$ . The value of n is _____.

JEE Advanced 2025

LEVELJEE Advanced

As shown in the figures, a uniform rod $O O^{'}$ of length $l$ is hinged at the point $O$ and held in place vertically between two walls using two massless springs of same spring constant. The springs are connected at the midpoint and at the top-end ( $O^{'}$ ) of the rod, as shown in Fig. 1 and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is $f_{1}$ . On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2 and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is $f_{2}$ . Ignoring gravity and assuming motion only in the plane of the diagram, the value of $\frac{f _{1}}{f _{2}}$ is:

2

\frac{5}{2}

\frac{2}{5}