The potential energy of a particle of mass 1 kg in motion along the x-axis is given by U = 4 (1 - cos 2x) J Here x is in metres . Find the period of small oscillations.

A body of mass m falls from a height h on to the pan of a spring balance. The masses of the pan and spring are negligible. The spring constant of the spring is k. Having stuck to the pan the body starts performing harmonic oscillations in the vertical direction. Find the amplitude and energy of oscillation.

Two linear simple harmonic motions of equal amplitudes and frequencies ω and 2ω are impressed on a particle along the axes of X and Y respective/yo If the initial phase difference between them is π/2 find the resultant path followed by the particle.

A particle is vibrating in SHM when the displacements of the particle from its equilibrium position are x₁ and x₂ it has velocities v₁ and v₂ respectively. Show that its time period is given by \(T=2\pi\sqrt{x_1^2-x_2^2\over v_2^2-v_1^2}\)

Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (ω is any positive constant):
(a) sin ωt – cos ωt
(b) sin3 ωt
(c) 3 cos(\(\frac{\pi}{4}\)-2ωt)
(d) cosωt + cos3ωt + cos5ωt
(e) exp(-ω²t²)
(f) 1 + ωt + ω²t²

The motion of a particle executing simple harmonic motion is described by the displacement function, x(t) = A cos (ωt + φ ).
If the initial (t = 0) position of the particle is 1 cm and its initial velocity is ω cm/s, what are its amplitude and initial phase angle ? The angular frequency of the particle is π s^–1. If instead of the cosine function, we choose the sine function to describe the SHM : x = B sin (ωt + α), what are the amplitude and initial phase of the particle with the above initial conditions.

Figures  correspond to two circular motions. The radius of the circle, the period of revolution, the initial position, and the sense of revolution (i.e. clockwise or anti-clockwise) are indicated on each figure.

Obtain the corresponding simple harmonic motions of the x-projection of the radius vector of the revolving particle P in each case.

Find the expression for time period of motion of a body suspended by two springs connected in parallel and series.

What do you understand by undamped and damped simple harmonic oscillations? Show that the time periods for vertical harmonic oscillations of the three systems shown in Figs. (a), (b) and (c) are in the ratio of \(1:\sqrt2:{1\over \sqrt2}.\) identical, each having a force constant k

Suppose a tunnel is dug through the earth from one side to the other side along a diameter. Show that the motion of a particle dropped into the tunnel is simple harmonic motion. Find the time period. Neglect all the frictional forces and assume that the earth has a uniform density. G = 6.67 x 10^-11 Nm² kg^-2; density of earth = 5.51 x 10³ kg m^-3