Question

"A satellite requires 88.5 min to orbit Earth once. Assume a circular orbit. 1) What is the circumference of the satellites orbit

Answer 1

circumference of the satellite orbit = 4.13 × 10⁷ m

Step-by-step explanation:

Given that:

the time period T = 88.5 min = 88.5 × 60 = 5310 sec

The mass of the earth
`M_e` = 5.98 × 10²⁴ kg

if the radius of orbit is r,

Then,

`(V^2)/(r) = (GM_e)/(r^2)`

`{V^2} = (GM_e r)/(r^2)`

`{V^2} = (GM_e )/(r)`

`{V} =\sqrt{ (GM_e )/(r)}`

Similarly :

`T = \sqrt{\frac{ 2 \pi r} {V} }`

where;
`{V} =\sqrt{ (GM_e )/(r)}`

Then:

`T = {\frac{ 2 \pi r^(3/2)} {\sqrt{ {GM_e }} }`

`5310= {\frac{ 2 \pi r^(3/2)} {\sqrt{ {6.674* 10^(-11) * 5.98 * 10^(24) }} }`

`5310= {\frac{ 2 \pi r^(3/2)} {\sqrt{ 3.991052 * 10^(14) }}`

`5310= {\frac{ 2 \pi r^(3/2)} {19977617.48}`

`5310 * 19977617.48= 2 \pi r^(3/2)}`

`1.06081149 * 10^(11)= 2 \pi r^(3/2)}`

`(1.06081149 * 10^(11))/(2 \pi)= r^(3/2)}`

`r^(3/2)} = (1.06081149 * 10^(11))/(2 \pi)`

`r^(3/2)} = 1.68833392 * 10^(10)`

`r= (1.68833392 * 10^(10))^(2/3)}`

`r= 2565.38^2`

r = 6579225 m

The circumference of the satellites orbit can now be determined by using the formula:

circumference = 2π r

circumference = 2π × 6579225 m

circumference = 41338489.85 m

circumference of the satellite orbit = 4.13 × 10⁷ m