Maharashtra Board Solutions for Chapter: Oscillations, Exercise 5: Exercises

Using differential equation of linear S.H.M, obtain the expression for acceleration in S.H.M

Draw graphs of displacement, velocity and acceleration against phase angle, for a particle performing linear S.H.M. from the positive extreme position. Deduce your conclusions from the graph.

The period of oscillation of a body of mass ${\mathrm{m}}_1$ suspended from a light spring is T. When a body of mass ${\mathrm{m}}_2$ is tied to the first body and the system is made to oscillate, the period is 2T. Compare the masses ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$

The displacement of an oscillating particle is given by $\mathrm{x}=\mathrm{asin}\mathrm{\omega t}+\mathrm{bcos}\mathrm{\omega t}$  where a, b and $\mathrm{\omega }$ are constants. Prove that the particle performs a linear S.H.M. with amplitude $\mathrm{A}=\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}$

Two parallel S.H.M.s represented by ${\mathrm{x}}_1=5\sin (4\mathrm{\pi t}+\mathrm{\pi }/3) \mathrm{cm}$  and ${\mathrm{x}}_2=3\sin$ $(4\mathrm{\pi t}+\mathrm{\pi }/4) \mathrm{cm}$ are superposed on a particle. Determine the amplitude and epoch of the resultant S.H.M.

A $20 \mathrm{cm}$ wide thin circular disc of mass $200 \mathrm{g}$ is suspended to a rigid support from a thin metallic string. By holding the rim of the disc, the string is twisted through $60°$and released. It now performs angular oscillations of period $1 \mathrm{second}$. Calculate the maximum restoring torque generated in the string under undamped conditions.$\left({\mathrm{\pi }}^3\approx 31\right)$

Find the number of oscillations performed per minute by a magnet is vibrating in the plane of a uniform field of $1.6\times {10}^{-5}\mathrm{Wb}/{\mathrm{m}}^2$. The magnet has moment of inertia $3\times {10}^{-6}{\mathrm{kgm}}^2$ and magnetic moment $3{\mathrm{Am}}^2$. 

A wooden block of mass m is kept on a piston that can perform vertical vibrations of adjustable frequency and amplitude. During vibrations, we don't want the block to leave the contact with the piston. How much maximum frequency is possible if the amplitude of vibrations is restricted to $25 \mathrm{cm}$? In this case, how much is the energy per unit mass of the block? $\left(\mathrm{g}\approx {\mathrm{\pi }}^2\approx 10{\mathrm{ms}}^{-2}\right)$