If a spring of stiffness $k$ is cut into two parts $A$ and $B$ of length $l_{A}: l_{B}=2: 3$, then the stiffness of spring $A$ is given by

A mass $m$ attached to a spring oscillates every $2\, sec$. If the mass is increased by $2 \,kg$, then time-period increases by $1\, sec$. The initial mass is ..... $kg$

A $2\, Kg$ block moving with $10\, m/s$ strikes a spring of constant $\pi ^2 N/m$ attached to $2\, Kg$ block at rest kept on a smooth floor. The time for which rear moving block remain in contact with spring will be ... $\sec$

Figure $(a)$ shows a spring of force constant $k$ clamped rigidly at one end and a mass $m$ attached to its free end. A force $F$ applied at the free end stretches the spring. Figure $(b)$ shows the same spring with both ends free and attached to a mass $m$ at etther end. Each end of the spring in Figure $( b )$ is stretched by the same force $F.$

$(a)$ What is the maximum extension of the spring in the two cases?

$(b)$ If the mass in Figure $(a)$ and the two masses in Figure $(b)$ are released, what is the period of oscillation in each case?

If a spring extends by $x$ on loading, then energy stored by the spring is (if $T$ is the tension in the spring and $K$ is the spring constant)

A mass $\mathrm{m}$ is suspended from a spring of negligible mass and the system oscillates with a frequency $f_1$. The frequency of oscillations if a mass $9 \mathrm{~m}$ is suspended from the same spring is $f_2$. The value of $\frac{f_1}{f_{.2}}$ is_____________.