$A$ simple pendulum of length $L$ has mass $m$ and it oscillates freely with amplitude $A$. At the extreme position,its potential energy is ($g =$ acceleration due to gravity).

If the temperature of a pendulum with time period $T$ is increased by $\Delta \theta$,the change in the time period of the pendulum is .......

$A$ bob of a simple pendulum of mass $m$ is displaced through $90^{\circ}$ from the mean position and released. When the bob is at the lowest position,the tension in the string is:

To show that a simple pendulum executes simple harmonic motion,it is necessary to assume that

$A$ simple pendulum of length $1 \,m$ is freely suspended from the ceiling of an elevator. The time period of small oscillations as the elevator moves up with an acceleration of $2 \,m/s^2$ is (use $g=10 \,m/s^2$).

$A$ cylindrical plastic bottle of negligible mass is filled with $310\, ml$ of water and left floating in a pond with still water. If pressed downward slightly and released,it starts performing simple harmonic motion at angular frequency $\omega$. If the radius of the bottle is $2.5\, cm$,then $\omega$ is close to ..... $rad\, s^{-1}$ (density of water $= 10^3\, kg/m^3$).