Question

4.
A satellite moving in a circular path of radius
'r' around earth has a time period T. If its
radius slightly increases by 4%, then
percentage change in its time period is
1) 1% 2) 6% 3) 3% ​

Answer 1

Answer:

A satellite moving in a circular path of radius

'r' around earth has a time period T. If its

radius slightly increases by 4%, then

percentage change in its time period is

1) 1% 2) 6% 3) 3%

Answer 2

Answer:

Time period increases by 6%.

(2) is correct option.

Explanation:

Given that,

Radius of circular path = r

Time Period = T

We need to calculate the percentage change in its time period

According to Kepler's law,

$T^2=kr^3$

On differentiating

$2T\times\dfrac{dT}{dr}=3\times kr^2$

$2T\times\dfrac{dT}{dr}=3\times\dfrac{T^2}{r}$

$\dfrac{dT}{dr}=\dfrac{3}{2}\times\dfrac{T}{r}$

$\Delta T=\dfrac{3}{2}\times\dfrac{T}{r}\times\Delta r$....(I)

If its  radius slightly increases by 4%.

$\Delta r=\dfrac{4}{100}r$

$r=0.04 r$

Put the value of $\Delta r$ in equation (I)

$\Delta T=\dfrac{3}{2}\times\dfrac{T}{r}\times0.04 r$

$\Delta T=0.06 T$

$\dfrac{\Delta T}{T}=0.06$

$\dfrac{\Delta T}{T}\times100=0.06\times100$

$\dfrac{\Delta T}{T}\times100=6\%$

Hence, Time period increases by 6%.