Question

A pendulum with a string of length 2 m has a period of about 2.8 s. What would be the period of the pendulum if the length of the string were increased to 8 m? Please answer this question with formula.


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Answer 1

Answer:

$\boxed{\mathfrak{Final \ time \ period \ of \ pendulum = 5.6 \ s}}$

Given:

Initial length of pendulum ($\rm l_1$) = 2 m

Final length of pendulum ($\rm l_2$) = 8 m

Initial time period of pendulum ($\rm T_1$) = 2.8 s

To Find:

Final time period of pendulum ($\rm T_2$)

Explanation:

Formula of time period of oscillation of simple pendulum:

$\boxed{ \bold{T = 2\pi \sqrt{ \dfrac{l}{g} } }}$

From the above formula we can say that time period is proportional to square root of the length of the pendulum i.e.

$\bold{T \propto \sqrt{l} }$

So,

$\rm \implies \dfrac{T_1}{T_2} = \sqrt{\dfrac{l_1}{l_2}} \\ \\ \rm \implies \dfrac{2.8}{T_2} = \sqrt{ \dfrac{2}{8} } \\ \\ \rm \implies \dfrac{2.8}{T_2} = \sqrt{ \dfrac{1}{4} } \\ \\ \rm \implies \dfrac{2.8}{T_2} = \dfrac{1}{2} \\ \\ \rm \implies T_2 = 2 \times 2.8 \\ \\ \rm \implies T_2 =5.6 \: s$

Answer 2

Answer :

➥ Final time period of a Pendulum = 5.6 sec

Given :

➤ Intial length of a Pendulum (l₁) = 2 m

➤ Final length of a Pendulum (l₂) = 8 m

➤ Intial time period of Pendulum (T₁) = 2.8 sec

To Find :

➤ Final time period of Pendulum (T₂) = ?

Solution :

For solving this question, let's first know about Time period.

• When the frequency of a wave increases, then the time period of the wave decreases.
• The Time period is denoted by T.
• The unit of time period is second.
• Time period and Frequency are in a reciprocal relationship.

We can find the Final time period of a Pendulum by using the formula:

❍ So let's calculate Final time period of Pendulum (T₂) !

As we know that

$\tt{ :\implies \dfrac{T_1}{T_2} = \sqrt{ \dfrac{l_1}{l_2} } }$

$\tt{ :\implies \dfrac{2.8}{T_2} = \sqrt{ \dfrac{2}{8} } }$

$\tt{ :\implies \dfrac{2.8}{T_2} = \sqrt{ \dfrac{1}{4} } }$

$\tt{ :\implies \dfrac{2.8}{T_2} = \dfrac{1}{2} }$

$\tt{ :\implies T_2 \times 1 = 2.8 \times 2}$

$\tt{ :\implies T_2 = 2 \times 2.8}$

$\bf{ :\implies \underline{ \: \: \underline{ \red{ \: \: T_2 = 5.6 \: sec \: \: }} \: \: }}$

║Hence, the final time period of a Pendulum is 5.6 sec.║