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dimaraw [331]
3 years ago
11

A satellite of mass m is in a circular orbit of radius R2 around a spherical planet of radius R1 made of a material with density

p. ( R2 is measured from the center of the planet, not its surface.) Use G for the universal gravitational constant. Part A Find the kinetic energy of this satellite, K.Part B Find U, the gravitational potential energy of the satellite. Take the gravitational potential energy to be zero for an object infinitely far away from the planet.
Physics
1 answer:
Amiraneli [1.4K]3 years ago
6 0

Answer:

a)      K = 2/3 π G m ρ R₁³ / R₂ ,  b) U = - G m M / r

Explanation:

The law of universal gravitation is

     F = G m M / r²

Part A

Let's use Newton's second law

     F = m a

The acceleration is centripetal

     a = v² / R₂

     

      G m M / R₂² = m v² / R₂

      v² = G M / R₂

They give us the density of the planet

    ρ = M / V

    V = 4/3 π R₁³

    M =   ρ V

    M =   ρ 4/3 π R₁³

    v² = 4/3 π G  ρ R₁³ / R₂

    K = ½ m v²

    K = ½ m (4/3 π G ρ R₁³ / R₂)

    K = 2/3 π G m ρ R₁³ / R₂

Part B

Potential energy and strength are related

     F = - dU / dr

     ∫ dU = - ∫ F. dr

The force was directed towards the center and the vector r outwards therefore there is an angle of 180º between the two cos 180 = -1

    U- U₀ = G m M ∫ dr / r²

    U - U₀ = G m M (- r⁻¹)

We evaluate for

    U - U₀ = -G m M (1 / r_{f} -  1 /r_{i})

They indicate that for ri = ∞     U₀ = 0

    U = - G m M / r

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Answer:

h=25.52\times10^6 m

Explanation:

Initial speed, v = 10 x 10^3 m/s

Mass of the earth, M = 6 x 10^24 kg

Radius of the earth, R = 6.4 x 10^6 m

Maximum from the surface of earth, h = ?

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Solution:

Potential energy at maximum height =  ( Potential + Kinetic energy ) at the surface

-G M m / ( R + h )=- G M m / R + (1/2) m v^2

- G M / ( R + h ) = - G M / R + (1/2) v^2

-2\times G M / ( R + h ) = ( - 2 G M / R ) + v^2

-2\times6.67\times10^{-11}\times6\times10^{24}/ ( R + h )

=( (- 2\times 6.67\times10^{-11}\times6\times10^{24}) /(6.4\times10^6)} +10000^2

=-2.50625\times10^7 J

=- 8\times10^{14} / ( R + h )=-2.50625\times 10^7

R+h=31.92\times10^{6}

h=31.92\times10^{6}-6.4\times10^6

h=25.52\times10^6 m

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