Question

compare the time periods of two pendulums of length 1m and 9m

Answer 1

see attachment......... .........

Answer 2

Given,

Two pendulums, of lengths 1 m and 9 m.

To find,

Comparison of the time periods of the two.

Solution,

A pendulum is a body or a weight that is suspended from a fixed point or pivot so that it could swing back and forth, freely.

The time period of a pendulum is given by,

$T=2\pi \sqrt{\frac{L}{g} }$

Where,

L = length of the string to which the weight is attached (or the pendulum's length)

g = acceleration due to gravity.

Now, for calculating the time periods, let the pendulums with lengths 1 m and 9 m be pendulum 1 and 2 respectively.

For pendulum 1,

$T_1=2\pi \sqrt{\frac{L_1}{g} }$

As L₁ = 1 m,

⇒ $T_1=2\pi \sqrt{\frac{1}{g} }$

⇒ $T_1=2\pi{\frac{1} {\sqrt{g} }$

For pendulum 2,

$T_2=2\pi \sqrt{\frac{L_2}{g} }$

As L₂ = 9 m,

⇒ $T_2=2\pi \sqrt{\frac{9}{g} }$

⇒ $T_2=2\pi{\frac{3} {\sqrt{g} }$

Now, for comparison, dividing both the time periods as,

$\frac{T_1}{T_2}= \frac{2\pi \frac{1}{\sqrt{g}} }{2\pi \frac{3}{\sqrt{g} } }$

Simplifying the above equation, we can clearly see that,

$\frac{T_1}{T_2}= \frac{1}{3}$

Or,

$T_1= \frac{T_2}{3}$

Therefore, the time period of the pendulum 1 will be one-third of the time period of the pendulum 2.