$A$ system is oscillating with undamped simple harmonic motion. Then the

Due to some force $F_1$,a body oscillates with a period of $4/5 \, s$,and due to another force $F_2$,it oscillates with a period of $3/5 \, s$. If both forces act simultaneously,the new period will be .... $s$.

Two pendulums with identical bobs and lengths are suspended from a common support such that in the rest position the two bobs are in contact. After being displaced by $5^o$,the bob $A$ is released from rest at $t = 0$. Subsequently,it collides elastically head-on with the other bob $B$. Identify the graph showing the variation in energy of pendulum $A$ with time for $0 \leqslant t \leqslant T$ (where $T$ is the period of either pendulum).

$A$ graph of the square of the velocity against the square of the acceleration of a given simple harmonic motion is

The center of a disk of radius $r$ and mass $m$ is attached to a spring of spring constant $k$,inside a ring of radius $R > r$ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring,without slipping. The spring can only be stretched or compressed along the periphery of the ring,following Hooke's law. In equilibrium,the disk is at the bottom of the ring. Assuming small displacement of the disc,the time period of oscillation of the center of mass of the disk is written as $T = \frac{2 \pi}{\omega}$. The correct expression for $\omega$ is ($g$ is the acceleration due to gravity):

$A$ body performs linear $S$.$H$.$M$. with amplitude $a$. When it is at a distance $\frac{a}{3}$ from the extreme position,the magnitude of velocity is $\frac{1}{3}$ times the magnitude of acceleration. The period of $S$.$H$.$M$. is: