A satellite is in a circular Earth orbit of radius r. The area A enclosed by the orbit depends on r2 because A=πr2. Determine how the following properties of the satellite depend on r:

(a) period,

(b) kinetic energy,

(c) angular momentum, and

(d) speed.

Short Answer

Expert verified
  1. T2r3
  2. K1r
  3. Lr
  4. Vc1r

Step by step solution

01

Listing the given quantities

The satellite is orbiting around the earth with a radius r

The area enclosed by the orbit A=πr2

02

Understanding the concept of gravitational force and centripetal force

Using gravitational force and centripetal force, we can find out the relation of r with T and speed. Using critical velocity in the kinetic energy equation, we can findtherelation between r and K. Usingthemoment of inertia and angular velocity; we will gettherelation between r and L.

Formula:

T2=2r3GmK=GMm2(r)F=GMmr2

CentripetalForce=mv2rmv2r=mrω2L=I=mr2

03

(a) Calculation of relation between r and T

For the orbiting satellite in a circular orbit, centripetal force is provided by the gravitational force of the earth.

mrω2=GMmr2

ω2=GMr3 (1)

ω=T ……………………………(2)

Using this in equation 1 and adjusting variables,

T2=2r3GmT2r3

04

(b) Calculation of relation between r and K

The velocity required to rotate the satellite in the orbit with radius r is the critical velocity

Vc=GMr

Using this in the kinetic energy equation,

K=12mVc2

K=GMm2(r)

K1r

05

(c) Calculation of relation between L and r

To findtherelation between L and r:

Angular momentum is given by

L=Iω

I is the moment of inertia and is the angular velocity.

I=mr2

L=mr2ω

L=mr2vr

From part b),

Vc1r

Lr21rr

Lr

06

(d) Calculation of relation between V and r

For the orbiting satellite in a circular orbit, centripetal force is provided bythegravitational force of the earth.

mVc2r=GMmr2

Vc2=GMrVc=GMrVc1r

We can find the relationship of r with different quantities using Newton’s law of gravitation and the formula for the respective quantities.

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