A 50-kg satellite circles the Earth in an orbit with a period of 120 min. What minimum energy is required to change the orbit to another circular orbit with a period of 180 min? (Earth: radius = 6.4 106 m, mass = 6.0 1024 kg)

Question

Physics

Erick Parrish

21 March, 17:34

A 50-kg satellite circles the Earth in an orbit with a period of 120 min. What minimum energy is required to change the orbit to another circular orbit with a period of 180 min? (Earth: radius = 6.4 106 m, mass = 6.0 1024 kg)

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Answers (1)

Know the Answer?

Queenie · Answer 1

Answer: 2.94*10^8 J

Explanation:

Using the relation

T^2 = (4π^2/GMe) r^3

Where v = velocity

r = radius

T = period

Me = mass of earth = 6*10^24

G = gravitational constant = 6.67*10^-11

4π^2/GMe = 4π^2 / [ (6.67x10^-11 x6.0x10^24) ]

= 0.9865 x 10^-13

Therefore,

T^2 = (0.9865 * 10^-13) * r^3

r^3 = 1 / (0.9865 * 10^-13) * T^2

r^3 = (1.014 x 10^13) * T^2

To find r1 and r2

T1 = 120min = 120*60 = 7200s

T2 = 180min = 180*60 = 10800s

Therefore,

r1 = [ (1.014 x 10^13) 7200^2]^ (1/3) = 8.07 x 10^6 m

r2 = [ (1.014 x 10^13) 10800^2]^ (1/3) = 10.57 x 10^6 m

Required Mechanical energy

= - GMem/2 [1/r2 - 1/r1]

= (6.67 x 10^-11 x 6.0 x 10^24 * 50) / 2 * [ (1/8.07 * 10^-6) - (1/10.57 * 10^-6) ]

= (2001 x 10^7) / 2 * (0.1239 - 0.0945)

= (1000.5 * 10^7) * 0.0294

= 29.4147 * 10^7 J

= 2.94 x 10^8 J.