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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Two identical balls A and B each of mass 0.1 kg are attached to two identical massless springs. The spring mass system is constrained to move inside a rigid smooth pipe bent in the form of a circle as shown in the figure. The pipe is fixed in a horizontal plane. The centres of the balls can move in a circle of radius 0.06 m. Each spring has a natural length of 0.06$\pi$ m and force constant 0.1N/m. Initially both the balls are displaced by an angle $\theta = \pi /6$ radian with respect to the diameter $PQ$ of the circle and released from rest. The frequency of oscillation of the ball B is
Equations ${y_1} = A\sin \omega t$ and ${y_2} = \frac{A}{2}\sin \omega t + \frac{A}{2}\cos \omega t$ represent $S.H.M.$ The ratio of the amplitudes of the two motions is
A potential is given by $V(x)=k(x+a)^2 / 2$ for $x < 0$ and $V(x)=k(x-a)^2 / 2$ for $x > 0$. The schematic variation of oscillation period $T$ for a particle performing periodic motion in this potential as a function of its energy $E$ is
A particle is executing $SHM$ about $y=0$ along $y$-axis. Its position at an instant is given by $y=(7 \,m )$ sin( $\pi f)$. Its average velocity for a time interval $0$ to $0.5 \,s$ is ........... $m / s$
A particle is executing $SHM$ with amplitude $A,$ time period $T,$ maximum acceleration $a_o$ and maximum velocity $v_0.$ Its starts from mean position at $t=0$ and at time $t,$ it has the displacement $A/2,$ acceleration $a$ and velocity $v$ then
A block of mass $m$ is at rest on an another block of same mass as shown in figure. Lower block is attached to the spring, then the maximum amplitude of motion so that both the block will remain in contact is