## The potential energy U of an object of mass m that is separated by a distance R from an object of mass M is given by U= -GMm/RA. What is the

Question

The potential energy U of an object of mass m that is separated by a distance R from an object of mass M is given by U= -GMm/RA. What is the kinetic energy K of the satellite?Express your answer in terms of the potential energy .
B. Find an expression for the square of the orbital period.

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5 months 2021-08-04T06:59:23+00:00 1 Answers 41 views 0

A)  K = ½ U / r
, B)     T² = (4π² / G M) r³

Explanation:

A) It asks us for the kinetic energy of the satellite and they give us the relation of the potential synergy

U = – G M m / r

the force can be found from this expression

F = – dU / dr

F = – G M m / r²

The kinetic energy is given by the formula

K = ½ m v²

let’s use Newton’s second law

F = m a

G M m / r² = m a

where the acceleration is centripetal

a = v² / r

G M r² = v² / r

v² = G M r³

we substitute the expression for the kinetic energy

K = ½ m v²

K = ½ m G M r³

in terms of potential energy is

K = ½ (G M m r²) 1 / r

K = ½ U / r

B) The period of the orbit

We write Newton’s second law

G M m / r² = m a

a = v² / r

in a circular orbit the speed of the velocity is constant

v = d / t

the longitude of the orbit is

d = 2π r

v = 2π r / T

we substitute

G Mm / r² = m (2πr / T)² / r

G M / r³ = 4π² / T²

T² = (4π² / G M) r³