Two springs with negligible masses and force constants $K_1 = 200\, Nm^{-1}$ and $K_2 = 160\, Nm^{-1}$ are attached to a block of mass $m = 10\, kg$ as shown in the figure. Initially,the block is at rest at the equilibrium position where both springs are neither stretched nor compressed. At time $t = 0$,a sharp impulse of $50\, Ns$ is given to the block with a hammer.

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 centers 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 a force constant of $0.1 \ N/m$. Initially,both 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:

At $t=0$,a particle executing $SHM$ with a time period $3 \ s$ is in phase with another particle executing $SHM$. The time period of the second particle is $T$ (less than $3 \ s$). If they are again in the same phase for the third time after $45 \ s$,then the value of $T$ is .... . (in $s$)

For a particle executing $S.H.M.$,where $x$ is the displacement from the equilibrium position,$v$ is the velocity at any instant,and $a$ is the acceleration at any instant,then:

You are holding a shallow circular container of radius $R$,filled with water to a height $h$ $(h \ll R)$. When you walk with speed $v$,it is seen that water starts spilling over. This happens due to the resonance of the periodic impulse given to the container (due to walking) with the oscillation of the water in the container. If the time period of water oscillating in the container is inversely proportional to $\sqrt{h}$,then $v$ is proportional to

$A$ horizontal board is performing simple harmonic oscillations horizontally with an amplitude of $0.3 \ m$ and a period of $4 \ s$. What is the minimum coefficient of friction between a heavy body placed on the board and the board itself if the body is not to slip?