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 $S.H.M.$ of time period $T$. If the mass is increased by m, the time period becomes $5T/3$. Then the ratio of $m/M$ is

A body of mass $0.01 kg$ executes simple harmonic motion $(S.H.M.)$ about $x = 0$ under the influence of a force shown below : The period of the $S.H.M.$ is …. $s$

Show that the oscillations due to a spring are simple harmonic oscillations and obtain the expression of periodic time.

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?

A steady force of $120\ N$ is required to push a boat of mass $700\ kg$ through water at a constant speed of $1\ m/s$ . If the boat is fastened by a spring and held at $2\ m$ from the equilibrium position by a force of $450\ N$ , find the angular frequency of damped $SHM$  ….. $rad/s$