Question

The Time Periods of two simple pendula,having different lengths,is the same on two different planets.If the length of two pendula are in Ratio 1:9 ,then the ratio of Acceleration due to gravity on the Two planets is?


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Answer 1

Answer :-

◾As provided :-

▪️Ratio of length of strings = 1:9

▪️Time period of both strings = T

◾Let the

▪️Length of string 1 be = x

▪️Gravitational pull on planet 1 = $g_1$

▪️Length of string 2 be = 9x

▪️Gravitational pull on planet 2 = $g_2$

◾Now first of all we should know how to find out the Time Period of simple pendulum .

◾Time Period of simple pendulum

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

◾Where

▪️T = Time Period

▪️L = Length of string

▪️g = Gravitational pull of the planet

◾Now on Planet 1

$T = 2\pi \sqrt{\dfrac{x}{g_1}}$

On squaring both sides :-

$\implies T^2 = 4\pi^2 \times \dfrac{x}{g_1}$

$\implies \dfrac{T^2}{4\pi^2(x)} = \dfrac{1}{g_1}$

$\implies \dfrac{4\pi^2(x)}{T^2} = g_1 \:\:\:...(i)$

◾Now on Planet 2

$T = 2\pi \sqrt{\dfrac{9x}{g_2}}$

On squaring both sides :-

$\implies T^2 = 4\pi^2 \times \dfrac{9x}{g_2}$

$\implies \dfrac{T^2}{4\pi^2(9x)} = \dfrac{1}{g_2}$

$\implies \dfrac{4\pi^2(9x)}{T^2} = g_2 \:\:\:...(ii)$

◾Now Ratio of Gravitation pull of Planet 1 : Planet 2

= equation (i) : equation (ii)

$= g_1 : g_2$

$= \dfrac{4\pi^2(x)}{T^2} : \dfrac{4\pi^2(9x)}{T^2}$

$= \dfrac{4\pi^2(x)}{T^2} \times \dfrac{T^2}{4\pi^2(9x)}$

$= \dfrac{1}{9}$

◾So

$\bold { g_1 : g_2 = 1:9}$

Answer 2

Answer:

The ratio of acceleration due to gravity on the two planets is 1 : 9.

Explanation:

By using the formula of time period of a simple pendulum.

T = 2π √l/g

where, 2π is the constant of proportionally.

Here, there is the time periods of two simple pendula,having different time period.

T1 = T2

So,

T1 = T2

=> 2π √l1/g1 = 2π √l2/g2

2π is cancelled.

Then,

=> √l1/g1 = √l2/g2

√ is cancelled.

So,

=> l1/g1 = l2/g2

.°. l1/l2 = g1/g2

Here, there is the lengths of two simple pendula,having different length.

=> l1 : l2 = 1 : 9

=> g1 / g2 = l1 / l2

We have to find g1 and g2.

=> g1 / g2 = l1 / l2

=> g1 / g2 = 1 / 9

.°. g1 : g2 = 1 : 9

Therefore, the ratio of acceleration due to gravity on the two planets is 1 : 9.