Question

The time period $T$ is found to depend upon $L$ as
(A) $T \propto L$
(B) $T \propto {L^2}$
(C) ${T^2} \propto L$
(D) $T \propto \dfrac{1}{L}$

Answer 1

The relation between the time period and the length is given from the formula of the time period of a pendulum. The time period of a pendulum is given by the formula, $T = 2\pi \sqrt {\dfrac{L}{g}}$ . So since the acceleration due to gravity is constant, so the time period is directly proportional to the square root of the length.

Complete step by step answer:
For a pendulum, the time period depends on the length of the string of the pendulum. This time period can be found out from the formula,
$\Rightarrow T = 2\pi \sqrt {\dfrac{L}{g}}$
Now, the acceleration due to gravity is taken approximately constant on the surface of the earth. So the value of the time period of the pendulum directly depends on the length of the pendulum.
Therefore taking the other variables as constant in the above equation, we have
$\Rightarrow T = A\sqrt L$ where $A$ is a constant whose value is given by $A = \dfrac{{2\pi }}{{\sqrt g }}$
Therefore we can remove the equality sign and have,
$\Rightarrow T \propto \sqrt L$
Now we can take squares on both sides of this equation. So we will have,
$\Rightarrow {T^2} \propto L$
So the square of the time period is directly proportional to the length of the pendulum.
Hence the option (B); ${T^2} \propto L$ is correct.

Note:
The time period of a pendulum is only dependent on the two factors, which are the length and the acceleration due to gravity. It is independent of the other factors such as the mass of the pendulum.
The square of the time period of a pendulum is directly proportional to the length where the length is the distance between the point of suspension and the center of mass of the body.