A mass $M$ is suspended from a spring of negligible mass. The spring is pulled a little and then released so that the mass executes simple harmonic oscillations with a time period $T$. If the mass is increased by m then the time period becomes $\left( {\frac{5}{4}T} \right)$. The ratio of $\frac{m}{{M}}$ is
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A mass $m$ is attached to two springs as shown in figure. The spring constants of two springs are $K _1$ and $K _2$. For the frictionless surface, the time period of oscillation of mass $m$ is
Two massless springs with spring constants $2\,k$ and $2\,k$, carry $50\, g$ and $100 \,g$ masses at their free ends. These two masses oscillate vertically such that their maximum velocities are equal. Then, the ratio of their respective amplitudes will be
A heavy small-sized sphere is suspended by a string of length $l$. The sphere rotates uniformly in a horizontal circle with the string making an angle $\theta $ with the vertical. Then the time period of this conical pendulum is
When a mass $m$ is hung from the lower end of a spring of neglibgible mass, an extension $x$ is produced in the spring. The time period of oscillation is
Starting from the origin a body oscillates simple harmonically with a period of $2\ s$. After what time will its kinetic energy be $75\%$ of the total energy?
Two simple harmonic motions $y_1 = A \sin \omega t$ and $y_2 =A \cos \omega t$ are superimposed on a particle of mass $m.$ The total mechanical energy of the particle is :
A particle of mass $m$ is dropped from a height $R$ equal to the radius of the earth above the tunnel dug through the earth as shown in the figure. Hence the correct statement is
Two particles $P$ and $Q$ start from origin and execute Simple Harmonic Motion along $X-$axis with same amplitude but with periods $3$ seconds and $6$ seconds respectively. The ratio of the velocities of $ P$ and $Q$ when they meet is
If a Second's pendulum is moved to a planet where acceleration due to gravity is $4$ times, the length of the second's pendulum on the planet should be made .......... times
Two simple harmonic motions are represented by the equations ${y_1} = 0.1\sin \left( {100\pi t + \frac{\pi }{3}} \right)$ and ${y_2} = 0.1\cos \pi t.$ The phase difference of the velocity of particle $1$ with respect to the velocity of particle $2$ is