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

At $t=0$,a particle executing $SHM$ with a time period $3 \ s$ is in phase with another particle executing $SHM$. The time period of the second particle is $T$ (less than $3 \ s$). If they are again in the same phase for the third time after $45 \ s$,then the value of $T$ is .... . (in $s$)

$A$ system of two identical rods ($L$-shaped) of mass $m$ and length $l$ are resting on a peg $P$ as shown in the figure. If the system is displaced in its plane by a small angle $\theta$,find the period of oscillations:

$A$ mass of $5\, {kg}$ is connected to a spring. The potential energy curve of the simple harmonic motion executed by the system is shown in the figure. $A$ simple pendulum of length $4\, {m}$ has the same period of oscillation as the spring system. What is the value of acceleration due to gravity on the planet where these experiments are performed? (In ${m} / {s}^{2}$)

Determine whether the following statements are True or False:
$1.$ The acceleration of $SHO$ at the mean position is maximum.
$2.$ The mechanical energy of $SHO$ depends on the maximum displacement.
$3.$ The periodic time for a seconds pendulum is $1 \, s$.
$4.$ If the frequency of $SHM$ is $v$,then the frequency of kinetic energy is also $v$.

Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system (see figure). Because of the torsional constant $k$,the restoring torque is $\tau = k\theta$ for an angular displacement $\theta$. If the rod is rotated by $\theta_0$ and released,the tension in it when it passes through its mean position will be