$A$ spring has a certain mass suspended from it and its period for vertical oscillation is $T$. The spring is now cut into two equal halves and the same mass is suspended from one of the halves. The period of vertical oscillation is now

Two identical point masses $P$ and $Q$,suspended from two separate massless springs of spring constants $k_1$ and $k_2$,respectively,oscillate vertically. If their maximum speeds are the same,the ratio $(A_Q / A_P)$ of the amplitude $A_Q$ of mass $Q$ to the amplitude $A_P$ of mass $P$ is

$A$ mass of $2 \, kg$ is attached to a spring with a spring constant of $50 \, N/m$. The block is pulled to a distance of $5 \, cm$ from its equilibrium position at $x = 0$ on a horizontal frictionless surface and released from rest at $t = 0$. Write the expression for its displacement at any time $t$.

$A$ block of mass $m$,attached to a spring of spring constant $k$,oscillates on a smooth horizontal table. The other end of the spring is fixed to a wall. The block has a speed $v$ when the spring is at its natural length. Before coming to an instantaneous rest,if the block moves a distance $x$ from the mean position,then

$A$ mass of $2.0\, kg$ is placed on a flat pan attached to a vertical spring fixed on the ground as shown in the figure. The mass of the spring and the pan is negligible. When pressed slightly and released,the mass executes simple harmonic motion. The spring constant is $200\, N/m$. What should be the minimum amplitude of the motion so that the mass gets detached from the pan? (Take $g = 10\, m/s^2$)

When a body of mass $8 \,kg$ is attached to a spring balance, the reading of the balance is $20 \,cm$. Instead of $8 \,kg$, if another body of mass $M$ is suspended from the spring balance and is made to oscillate vertically, the time period of oscillation is $\frac{\pi}{5} \,s$, then the value of $M$ is (Acceleration due to gravity $= 10 \,m/s^2$) (in $\,kg$)