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 .... $s$.

For a particle executing simple harmonic motion, match the following statements (conditions) from Column-$I$ to statements (shapes of graph) in Column-$II$.

Column-$I$	Column-$II$
$(A)$ Velocity-displacement graph $(\omega \neq 1)$	$(i)$ Straight line
$(B)$ Acceleration-displacement graph	$(ii)$ Sinusoidal
$(C)$ Acceleration-time graph	$(iii)$ Circle
$(D)$ Acceleration-velocity graph $(\omega \neq 1)$	$(iv)$ Ellipse

$A$ mass $m$ oscillates with simple harmonic motion with frequency $f = \frac{\omega}{2\pi}$ and amplitude $A$ on a spring with constant $K$. Therefore:

Column $I$ gives a list of possible sets of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column $II$. Match the set of parameters given in Column $I$ with the graph given in Column $II$.

Column $I$	Column $II$
$(A)$ Potential energy of a simple pendulum ($y$-axis) as a function of displacement ($x$-axis)	$(p)$ Parabolic curve opening upwards
$(B)$ Displacement ($y$-axis) as a function of time ($x$-axis) for a one-dimensional motion at zero or constant acceleration	$(q)$ Linear graph passing through origin
$(C)$ Range of a projectile ($y$-axis) as a function of its velocity ($x$-axis) when projected at a fixed angle	$(r)$ Linear graph with non-zero intercept
$(D)$ The square of the time period ($y$-axis) of a simple pendulum as a function of its length ($x$-axis)	$(s)$ Parabolic curve opening upwards (starting from origin)

As shown in the figures,a uniform rod $OO^{\prime}$ of length $l$ is hinged at the point $O$ and held in place vertically between two walls using two massless springs of same spring constant $K$. The springs are connected at the midpoint and at the top-end $(O^{\prime})$ of the rod,as shown in Fig. $1$,and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is $f_1$. On the other hand,if both the springs are connected at the midpoint of the rod,as shown in Fig. $2$,and the rod is made to oscillate by a small angular displacement,then the frequency of oscillation is $f_2$. Ignoring gravity and assuming motion only in the plane of the diagram,the value of $\frac{f_1}{f_2}$ is:

The maximum velocity and maximum acceleration of a particle performing a linear $S.H.M.$ are $\alpha$ and $\beta$ respectively. Then the path length of the particle is