A particle is subjected to two simple harmonic otions in the same direction having equal magnitude and equal frequency. If the resultant amplitude is equal to the amplitude of the individual motion. Fnd the phase difference between two individual motions.

A particle moving with SHM ina stright line has a speed of 6 m/s when 4 m/s from the centre of oscillation and a speed of 8 m/s when 3 m from the oscillation and the shortest time taken by the partile in moving from the extreme position to a point mid way between the extreme position and the centre.

A spring of force constant \({ 1200\ Nm }^{ -1 }\)is mounted on a horiontal table as a mass of 3 kg is attached to the free end of the spring, pulled sideways to adistance of 2.0 cm and released. Determine
(i) The frequency of oscillations.
(ii) The maximum acceleration of the mass, and
(iii) the maximum speed of the mass?

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.

A cylindrical piece of cork of density of base area A and height h floats in a liquid of density ρ_l . The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period \(T=2\pi \sqrt { \frac { h\rho }{ { \rho }_{ 1 }g } } \) where ρ is the density of cork. (Ignore damping due to viscosity of the liquid).

An air chamber of volume V has a neck area of cross section a into which a ball of mass m just fits and can move up and down without any friction (Fig.). Show that when the ball is pressed down a little and released, it executes SHM. Obtain an expression for the time period of oscillations assuming pressure-volume variations of air to be isothermal.

Show that for a particle in linear SHM the average kinetic energy over a period of oscillation equals the average potential energy over the same period.

Define simple harmonic motion (5HM). Two masses mI and m₂ are suspended together by a massless spring of spring constant k. When the masses are in equilibrium, mI is removed without disturbing the system. Find the angular frequency and amplitude oscillation of m₂

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