The time period of a simple pendulum in air is $T$. If the pendulum is immersed in water and executes $SHM$,its time period is $t$. The value of $\frac{T}{t}$ is [density of the bob is $\frac{5000}{3} \ kg \ m^{-3}$ and density of water is $1000 \ kg \ m^{-3}$].

$A$ pendulum clock is placed inside an elevator. Under what condition does it show the correct time?

$A$ $metre$ scale is suspended vertically from a horizontal axis passing through one end of it. Its time period would be ....... $\sec$

$A$ simple pendulum is taken to a place where its distance from the Earth's surface is equal to the radius of the Earth. Calculate the time period of small oscillations if the length of the string is $4.0 \ m$. (Take $g = \pi^2 \ m/s^2$ at the surface of the Earth.) (in $s$)

Two pendulums begin to swing simultaneously. If the ratio of the frequency of oscillations of the two is $7 : 8$,then the ratio of lengths of the two pendulums will be

$A$ simple pendulum of length $L$ is constructed from a point object of mass $m$ suspended by a massless string attached to a fixed pivot point. $A$ small peg is placed at a distance $2L/3$ directly below the fixed pivot point so that the pendulum would swing as shown in the figure. The mass is displaced $5$ degrees from the vertical and released. How long does it take to return to its starting position?