A mass $m$ is vertically suspended from a spring of negligible mass; the system oscillates with a frequency $n$. What will be the frequency of the system if a mass $4 m$ is suspended from the same spring

A spring having with a spring constant $1200\; N m ^{-1}$ is mounted on a hortzontal table as shown in Figure A mass of $3 \;kg$ is attached to the free end of the spring. The mass is then pulled sideways to a distance of $2.0 \;cm$ and released 

let us take the position of mass when the spring is unstreched as $x=0,$ and the direction from left to right as the positive direction of $x$ -axis. Give $x$ as a function of time $t$ for the oscillating mass if at the moment we start the stopwatch $(t=0),$ the mass is

$(a)$ at the mean position,

$(b)$ at the maximum stretched position, and

$(c)$ at the maximum compressed position. In what way do these functions for $SHM$ differ from each other, in frequency, in amplitude or the inittal phase?

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 particle at the end of a spring executes simple harmonic motion with a period ${t_1}$, while the corresponding period for another spring is ${t_2}$. If the period of oscillation with the two springs in series is $T$, then

A uniform stick of mass $M$ and length $L$ is pivoted at its centre. Its ends are tied to two springs each of force constant $K$ . In the position shown in figure, the strings are in their natural length. When the stick is displaced through a small angle $\theta $ and released. The stick