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 $v$.

(A) $1.$ False. At the mean position,the displacement $x = 0$,so acceleration $a = -\omega^2 x = 0$. The acceleration is zero at the mean position and maximum at the extreme positions.
$2.$ True. The total mechanical energy of an $SHO$ is given by $E = \frac{1}{2} k A^2$,where $A$ is the maximum displacement (amplitude). Thus,it depends on the square of the amplitude.
$3.$ False. $A$ seconds pendulum is defined as a pendulum whose period of oscillation is $2 \, s$ (one second for each swing).
$4.$ False. If the frequency of $SHM$ is $v$,the kinetic energy oscillates as $K = \frac{1}{2} k A^2 \cos^2(\omega t) = \frac{1}{4} k A^2 (1 + \cos(2\omega t))$. Since the angular frequency is $2\omega$,the frequency of kinetic energy is $2v$.