Exams › JEE Main › Physics

Three equal charges Q are fixed at the vertices of an equilateral triangle of side a, and a fourth charge q (of mass m) sits at the centroid in equilibrium. For small displacements of the fourth particle along the plane, what is the period of its oscillation?

1. 2*pi*sqrt((4*pi*e0*m*a³)/(9*sqrt(3)*Q*q))
2. pi*sqrt((4*pi*e0*m*a³)/(9*sqrt(3)*Q*q))
3. 2*pi*sqrt((2*pi*e0*m*a³)/(27*sqrt(3)*Q*q))
4. 2*pi*sqrt((pi*e0*m*a³)/(27*sqrt(3)*Q*q))

Correct answer: 2*pi*sqrt((4*pi*e0*m*a³)/(9*sqrt(3)*Q*q))

Solution

The centroid is at distance r = a/sqrt(3) from each vertex. For a small in-plane displacement the three Coulomb forces give a net linear restoring force F = -k*x with k = 9*sqrt(3)*Q*q/(4*pi*e0*a³) (after summing the radial-component derivatives of the three interactions and using r³ = a³/(3*sqrt(3))). The SHM period is T = 2*pi*sqrt(m/k) = 2*pi*sqrt((4*pi*e0*m*a³)/(9*sqrt(3)*Q*q)).

Related JEE Main Physics questions

• A block of mass M is attached to a horizontal spring and oscillates in simple harmonic motion with amplitude A1. When the block crosses the equilibrium position, a smaller mass m is gently placed on it, and thereafter the two move together. The new amplitude is A2. What is the value of the ratio A1/A2?
• An ideal gas is trapped in a vertical cylinder of cross-sectional area A by a frictionless piston of mass M. When the gas volume is V0 and its pressure is P0, the piston is in equilibrium. If the piston is given a small displacement from this position and then let go, and the entire setup is thermally isolated, the piston performs SHM with frequency
• A mass m is connected to a spring with spring constant k, giving it a natural angular frequency ω0. If an external driving force varying as cos(ωt) acts on the system, the oscillator’s displacement will be proportional to
• A particle undergoing simple harmonic motion has kinetic energy K0. The maximum potential energy and the total energy are, respectively,
• A simple pendulum is suspended from the roof of a lift that is initially at rest, and its period is T. The lift then moves upward such that the distance it covers is given by y = t², where y is in metres and t is in seconds. Taking g = 10 m/s², what will be the new time period of the pendulum?
• A damped oscillator’s amplitude falls to 0.9 of its initial value in 5 s. After an additional 10 s, what fraction of the initial amplitude will it have, if the factor is denoted by α?

⚔️ Practice JEE Main Physics free + battle 1v1 →