Question

The mass and radius of a planet are double that of the earth. If the time period of a simple pendulum on the earth is T. The time period on the planet is

Answer 1

We know that a simple pendulum consists of a mass m hanging from a string of length L and fixed at a pivot point P. When displaced to an initial angle and released, the pendulum will swing back and forth with periodic motion. The time for one complete cycle, a left swing and a right swing, is called the period. The period depends on the length of the pendulum and also to a slight degree on the amplitude, the width of the pendulum's swing. Pendulums are used in many engineered objects, such as clocks, metronomes, amusement park rides and earthquake seismometers. In addition, engineers know that understanding the physics of how pendulums behave is an important step towards understanding motion, gravity, inertia and centripetal force.

Complete step by step answer
We know that,
Time period of simple pendulum $=2 \pi \sqrt{\dfrac{\mathrm{L}}{\mathrm{g}}}$ Where $\mathrm{L}=$ Length of wire $/$ thread $\mathrm{g}=$ gravity of planet $\mathrm{g}=\dfrac{\mathrm{GM}}{\mathrm{R}^{2}}$
where
$\mathrm{G}=$ Gravitational constant
$\mathrm{M}=$ Mass of planet $\mathrm{R}=$ Radius of planet
$\mathrm{M}_{\mathrm{P}}=2 \mathrm{Me}$
$\mathrm{R}_{\mathrm{P}}=2 \mathrm{Re}$
where, $\mathrm{P}_{\mathrm{e}}$ stands for planet earth $\mathrm{T}_{\mathrm{P}} \propto \dfrac{1}{\sqrt{\mathrm{gP}}} \propto \sqrt{\dfrac{\mathrm{R}^{2}}{\mathrm{M}}} \propto \dfrac{\mathrm{R}}{\sqrt{\mathrm{M}}}$
$\Rightarrow \mathrm{T}_{\mathrm{P}}=\dfrac{2}{\sqrt{2}} \mathrm{T}_{\mathrm{e}}=\sqrt{2} \mathrm{T}_{\mathrm{e}}$

Time period on planet is $\sqrt{2}$ times time period on earth

Note Thus, we can say that the pendulum, body suspended from a fixed point so that it can swing back and forth under the influence of gravity. Pendulums are used to regulate the movement of clocks because the interval of time for each complete oscillation, called the period, is constant. Pendulum, body suspended from a fixed point so that it can swing back and forth under the influence of gravity. Pendulums are used to regulate the movement of clocks because the interval of time for each complete oscillation, called the period, is constant. Thus, it is a hypothetical apparatus consisting of a point mass suspended from a weightless, frictionless thread whose length is constant, the motion of the body about the string being periodic and, if the angle of deviation from the original equilibrium position is small, representing simple harmonic motion.