Question

The time period of a simple pendulum of infinite length is:
(a). $T=2\pi \sqrt{\dfrac{{{R}_{e}}}{g}}$
(b). Infinity
(c). $T=\pi \sqrt{\dfrac{{{R}_{e}}}{g}}$
(d). zero

Answer 1

Hint: As the length is infinite we can consider the length as the radius of the earth, then apply the formula for the time period (T) for the pendulum, take g=9.8m/s. The radius of the earth is 6400Km.

Complete step-by-step solution -
Taking Re (radius of the earth), as the length of the simple pendulum,
We know,
${{R}_{e}}$= $l$ = 6400Km
The formula for the time period of the pendulum is,
$T=2\pi \sqrt{\dfrac{l}{g}}$ ,l is the length of the pendulum, g is the acceleration due to gravity
 i.e,=9.8m/${{s}^{2}}$
Now substituting l with the ${{R}_{e}}$,(because we have already considered the radius of earth ${{R}_{e}}$ as the length of the pendulum).

T=$2\pi \sqrt{\dfrac{{{R}_{e}}}{9.8}}$
T=$3.14\times 2\sqrt{\dfrac{6400}{9.8}}$(g=9.8m/s$^{2}$ )
T=160.48 s
Therefore, the correct option is option (a).
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Additional Information:
 Time period is the time taken by the pendulum to finish one full oscillation.
Acceleration due to gravity is the free fall acceleration of an object in vacuum without any drag. This steady gain in the velocity is caused exclusively by the force of gravitational pull.
The amplitude of a pendulum is a measure of its change in a single period. There are various definitions of amplitude, which are all functions of the magnitude of the differences between the variable's extreme values.

Note: Whenever the length is infinite take the length as the radius of earth, consider ‘g’ as 9.8 always unless it's told to take it as 10,always try to find the value of T,while substituting the value always try to check it twice.