Question

Q28. A pendulum oscillates 81 times in 9 seconds. Find its time period and
frequency.​

Answer 1

$\large\bf{ \purple{ \underline{ \underline{Answer :}}}}$

• Time period of the pendulum is 0.1 seconds.

• Frequency of the pendulum is 10 Hz.

Given :

• Time taken by the pendulum to oscillate 81 times = 9 seconds.

• Number of oscillations = 81 times.

To Find :

• Time period and frequency of the pendulum.

Calculation :

Firstly we'll calculate the time period of the pendulum,

As we know that,

$\boxed{ \pmb{ \rm{ \red{Time \: period = \dfrac{Time \: taken}{Number \: of \: oscillations}}}}}$

Here,

• Time taken (T) = 9 seconds
• Number of oscillations = 81 times

Putting the values,

$\longrightarrow \sf {Time \: period}_{(Pendulum)} = \dfrac{Time \: taken}{Number \: of \: oscillations} \\ \\ \\ \longrightarrow \sf {Time \: period}_{(Pendulum)} = \cancel \dfrac{9 \: }{81} \\ \\ \\ \longrightarrow {\boxed{\sf {Time \: period}_{(Pendulum)} = 0.1 \: s}} \: \blue{ \bigstar}$

Therefore, time period of the pendulum is 0.1 seconds.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\dag$ Now, let's calculate the frequency for the pendulum,

Formula to be used :

• $\boxed{ \rm{ \pmb{ \pink{ \: Frequency \: (F) = \dfrac{1}{Time \: period \: (T) \: }}}}}$

$\bullet$ Frequency is always calculated in Hz.

Putting the values,

$\longrightarrow \sf \: Frequency = \cancel\dfrac{1}{0.1} \\ \\ \\ \longrightarrow { \boxed{\sf \: Frequency = 10 \: Hz}} \: \purple{ \bigstar}$

Therefore, frequency of the pendulum is 10 Hz.

Answer 2

$\huge\bf\underline\mathfrak\red{Answer :}$

1. $\text{Time period of pendulum}$ = $\sf\purple{0.1 \: seconds.}$
2. $\text{Frequency of pendulum}$ = $\sf\purple{10 \: Hertz.}$

$\huge\bf\underline\mathfrak\red{Step \: by \: step \: explanation :}$

$\huge\bf\underline\mathfrak\red{Given :}$

1. $\text{Number of oscillations by pendulum}$ = $\sf\purple{81 \: times.}$
2. $\text{Time taken for completing 81 oscillations.}$ = $\sf\purple{9 \: seconds}$

$\huge\bf\underline\mathfrak\red{To \: find :}$

• $\text{Time period and frequency of the pendulum.}$

$\huge\bf\underline\mathfrak\red{Solution :}$

$\sf\underbrace{Finding \: time \: period \: :- }$

$\underline\text{We have the formula for finding time period :-}$

⠀⠀⠀⠀⠀⠀$\color{green}\star\:\tt Time \: period \: = \dfrac{Number \: of \: oscillations}{Time \: taken}$

$\underline\text{Putting the given values :-}$

⠀⠀⠀⠀⠀⠀$\star\:\tt Time \: period \: = \dfrac{9}{81}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀$\implies$ $\sf\red{0.1 \: seconds}$

$\sf\underbrace{Finding \: Frequency \: :- }$

$\underline\text{We have the formula for finding frequency :-}$

⠀⠀⠀⠀⠀⠀$\color{green}\star\:\tt Frequency \: = \dfrac{1}{Time \: period}$

$\underline\text{Putting the calculated value :-}$

⠀⠀⠀⠀⠀⠀$\star\:\tt Frequency \: = \dfrac{1}{0.1}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀$\implies$ $\sf\red{10 \: Hz}$

Hence,

1. $\text{Time period = 0.1 seconds.}$
2. $\text{Frequency = 10 hertz.}$

$\huge\bf\underline\mathfrak\red{Learn \: More :}$

• When a pendulum completes it's one oscillation complete in a particular time, we call it as the time period of a pendulum.
• The time period of a pendulum depends upon length of the pendulum - How long is the pendulum or much longer is the string at which the ball is attached.
• Frequency of a pendulum means that during a particular given time, how much the pendulum oscillates or how many times the pendulum swing in to-and-fro motion.