$A$ simple pendulum of string length $30 \ cm$ performs $20$ oscillations in $10 \ s$. The length of the string required for the pendulum to perform $40$ oscillations in the same time duration is . . . . . . $cm$. [Assume that the mass of the pendulum remains same.]

$A$ trolley of mass $m_1$ is placed on a horizontal rigid pair of rails at the same height. $A$ mass $m_2$ is suspended from the trolley vertically by means of an ideal massless rope. The rope hangs between the rails without touching them. The trolley can move along smooth rails but cannot move in any other direction. The suspended mass is given a small oscillation and performs $SHM$ after being displaced slightly from its stable equilibrium position in two ways: first,perpendicular to the rails,and second,parallel to the rails. The ratio of the time periods of these two $SHM$s (second case to first case) is:

Time period of a simple pendulum is $T$ inside a lift when the lift is stationary. If the lift moves upwards with an acceleration $g / 2$,the time period of the pendulum will be

$A$ small sphere of radius $r$ is placed on a concave surface of radius of curvature $R$ a little away from the center. When the sphere is released,it oscillates. Assuming the oscillation to be simple harmonic motion and $r << R$,then the time period is

$A$ simple pendulum of length $L$ and mass (bob) $M$ is oscillating in a plane about a vertical line between angular limits $-\varphi$ and $+\varphi$. For an angular displacement $\theta$ $(|\theta| < \varphi)$,the tension in the string and the velocity of the bob are $T$ and $v$ respectively. Which of the following relations hold good under the above conditions?

$A$ string of a pendulum of length $l$ is displaced through $90^{\circ}$ from its vertical and released. Then,the minimum strength of the string needed to withstand the tension as the pendulum passes through its mean position is