$A$ spring has length $L$ and force constant $K$. It is cut into two springs of length $L_1$ and $L_2$ such that $L_1 = N L_2$ ($N$ is an integer). The force constant of the spring of length $L_1$ is:

To the free end of a spring hanging from a rigid support, a block of mass $m$ is hung and slowly allowed to come to its equilibrium position. Then the stretching in the spring is $d$. If the same block is attached to the same spring and allowed to fall suddenly, the amount of maximum stretching is: (force constant, $k$)

One-fourth length of a spring of force constant $K$ is cut away. The force constant of the remaining spring will be

$A$ bead of mass $m = 100 \,g$ is attached to one end of a spring of natural length $L$ and spring constant $k = \frac{(\sqrt{3}+1) mg}{L}$. The other end of the spring is fixed at point $A$ on a smooth vertical ring of radius $R$. The bead is at point $B$ such that the spring makes an angle of $30^{\circ}$ with the horizontal diameter. The normal reaction at $B$ just after it is released to move is (take $g = 9.8 \,ms^{-2}$): (in $\,N$)

What is the acceleration of $3 \,kg$ mass when the acceleration of $2 \,kg$ mass is $2 \,m/s^2$ as shown in the figure? (in $m/s^2$)

$A$ mass of $10 \ kg$ is suspended from a spring balance. It is pulled aside by a horizontal string so that it makes an angle of $60^{\circ}$ with the vertical. The new reading of the balance is