Question

a. If the pendulum has a length of 49 units, what is its period?
b. If the length of a pendulum is quadrupled, what happens to its
period?​

Answer 1

Answer:

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Answer 2

Given:

Length of the pendulum = $49units$

To be found:

a. The period of pendulum if it's length is $49units$.

b. What happens to it's period when it's length is quadrupled.

Formula to be used:

The period of a pendulum is given by:

$T=2\pi \sqrt{L/g}$ .......(1)

Where, $T$ = Period of the pendulum ($seconds$)

$L$ = Length of the pendulum (Any units of length)

$g$ = Acceleration due to gravity (Considering, $g=10m/s^{2}$ here)

Solution:

a. As per the given data, the length of the pendulum is $49units$.

Consider the period of pendulum for this length as '$T_1$'.

Now, for this length, the period of the pendulum is calculated using the equation (1) in the following way:

$T_1=2\pi \sqrt{L/g} \\T_1=2\pi \sqrt{49/10}\\ T_1=2\pi \sqrt{4.9}\\ T_1=13.91seconds$

⇒ The period of pendulum with length $49units$ is $13.91seconds$.

b. Now, in the second case, given that the length of the pendulum is quadrupled.

⇒ The length of the pendulum is increased by $4times$.

⇒ $L=4*49\\L=196units$

• Consider the period for this length as '$T_2$'.
• For this length, the period is calculated as follows:

$T_2=2\pi \sqrt{196/10}\\ T_2=2\pi \sqrt{19.6}\\ T_2=27.82seconds$

⇒ The period of the pendulum, when it's length is quadrupled is $27.82seconds$.

• Now, compare the two periods of the pendulum to find what happened to the period of pendulum when its length is quadrupled. For the comparison, divide the period '$T_2$' with the period '$T_1$'.

$T_2/T_1=27.82/13.91\\T_2/T_1 = 2$

⇒$T_2=2T_1$

⇒ The period of pendulum increases by $two$ times, if it's length is quadrupled.

Note:

The second part of the problem can be directly checked by using the formula mentioned in equation (1), by denoting periods and lengths as in the solution given above.

$T_2/T_1=(2\pi \sqrt{L_2/g})/(2\pi \sqrt{L_1/g})\\ T_2/T_1=(\sqrt{L_2}/\sqrt{g})/(\sqrt{L_1}/\sqrt{g})\\ T_2/T_1=\sqrt{L_2} /\sqrt{L_1}$........(2)

As per the given data, $L_2=4L_1$

Now, substitute this in equation (2).

$T_2/T_1=\sqrt{4L_1} /\sqrt{L_1}\\ T_2/T_1=\sqrt{4}\\ T_2/T_1=2\\T_2=2T_1$

Final Answer:

∴ a. The period of pendulum when it's length is $49units$ is $13.91 seconds$.

b. The period of pendulum increases by $two$ times, when it's length is quadrupled.