Question

If the length of the pendulum is made $9$ times and mass of the bob is made $4$ times, then the value of time period becomes:
A) $3T$
B) $\dfrac{3T}{{2}}$
C) $4T$
D) $2T$

Answer 1

Hint: The period of the pendulum only depends on the length of the string and it is independent of the mass of the bob. A simple pendulum with a longer string can have a long time period. There are two forces acting on the pendulum. They are forces of gravity and the tension force.

Formula Used:
We will be using the formula of the time period of a simple pendulum is $T = 2\pi \sqrt {\dfrac{l}{g}}$.

Complete step by step answer:
It is given that the length of the pendulum is $9$ times and the mass of the bob is $4$ times. To find the value of the period of time.
A simple pendulum consists of a string which is tied at one end to a pivot point and another end is attached to the bob of mass m. The period of a simple pendulum is the time taken to complete one full to-and fro swing. The length of the pendulum is nothing but it is the vertical distance between the Point of suspension and the center of the bob. It is represented by the letter ‘$l$’.
The periodic time of a simple pendulum is given by,
$T = 2\pi \sqrt {\dfrac{l}{g}} $
Where, g is the acceleration due to gravity, $l$ is the length of the string attached to the bob. The period of time is inversely proportional to the gravity. Hence the stronger the gravitational acceleration, the smaller the period of time.
${T_1} = T,$ ${l_1} = 1$, ${l_1} = 9l$
$\dfrac{{{T_1}}}{{{T_2}}} = \sqrt {\dfrac{{{l_1}}}{{{l_2}}}} = \sqrt {\dfrac{1}{9}} = \dfrac{1}{3}$
${T_2} = 3{T_1} = 3T$
If the length of the pendulum is increased by $9$ times, then the time becomes $3$ times.

Hence the correct option (A), $3T$.

Note: The pendulums are used to regulate the movement of clocks because the interval of time for each complete oscillation is called the period and that is constant. A simple pendulum is a mechanical arrangement which demonstrates the periodic motion.