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

Two pendulums begin to swing simultaneously. The first pendulum makes $11$ full oscillations when the other makes $9$. The ratio of the lengths of the two pendulums is

$A$ pendulum has a time period $T$ in air. When it is made to oscillate in water, its time period is $\sqrt{2} T$. Then the relative density of the material of the bob of the pendulum is (neglect damping).

$A$ simple pendulum hangs from the ceiling of a car. If the car accelerates with a uniform acceleration,the frequency of the simple pendulum will

The ratio of the period of oscillation of a conical pendulum to that of a simple pendulum is: (Assume the strings are of the same length in both cases and $\theta$ is the angle made by the string with the vertical in the case of the conical pendulum.)

$A$ pendulum clock is set to give correct time at sea level. This clock is moved to a hill station at an altitude of $2500 \ m$ above sea level. In order to keep correct time at the hill station,the length of the pendulum: