$A$ 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

Two bodies of masses $1\, kg$ and $4\, kg$ are connected to a vertical spring,as shown in the figure. The smaller mass executes simple harmonic motion of angular frequency $25\, rad/s$,and amplitude $1.6\, cm$ while the bigger mass remains stationary on the ground. The maximum force exerted by the system on the floor is ..... $N$ (take $g = 10\, m/s^2$).

$A$ mass $M$,attached to a horizontal spring,executes $S.H.M.$ with amplitude $A_1$. When the mass $M$ passes through its mean position,a smaller mass $m$ is placed over it and both of them move together with amplitude $A_2$. The ratio of $\frac{A_1}{A_2}$ is

$A$ spring balance has a scale that reads from $0$ to $50\; kg$. The length of the scale is $20\; cm$. $A$ body suspended from this balance,when displaced and released,oscillates with a period of $0.6\; s$. What is the weight of the body in $N$?

Define the torsional constant for a spring.

An object of mass $2 \,kg$ is attached to a spring with spring constant $8 \,N/m$. If the object is executing simple harmonic motion, then the number of cycles it completes in $66 \,s$ is