__Question 4__

A satellite of mass *m*S is in a circular orbit of radius *x
*about the Earth.

The Earth may be
considered to be an isolated uniform sphere with its mass *M
*concentrated at its centre.

**(a)
(i) **Show that the kinetic energy *E*K of the satellite is given by the expression

*E*K = *GMm*S / 2*x*

where *G
*is the gravitational constant. Explain your working. [3]

**(ii)
**State an expression, in terms of *G*, *M*, *m*S and *x*, for the potential
energy *E*P of
the

satellite. [1]

**(iii)
**Using answers from **(i) **and **(ii)**,
derive an expression for the total energy *E*T of the

satellite. [2]

**(b)
**Small resistive forces acting on the satellite cause the radius of
its circular orbit to change.

Use your answers in **(a)
**to state, for the satellite, whether each of the following
quantities

increases, decreases
or remains constant.

**(i)
**total energy [1]

**(ii)
**radius of orbit [1]

**(iii)
**potential energy [1]

**(iv)
**kinetic energy [1]

**Reference:** *Past Exam Paper – November 2015 Paper 41 &42 Q1*

**Solution:**

The
gravitational force provides the centripetal force.

{Gravitational
force = Centripetal force}

GMmS / x^{2} = mSv^{2} / x

{m_{s}v^{2}
= GMm_{s} / x

Multiply
by half on both sides,

½ m_{s}v^{2}
= GMm_{s} / 2x

E_{K}
= GMm_{s} / 2x}

**(ii)
**EP = – GMmS / x

**(iii)
**

ET = EK + EP

ET = GMmS / 2x – GMmS / x

ET = – GMmS / 2x

**(b)
**

**(i)
**decreases

{The total
energy decreases as work needs to be done against the resistive forces.}

**(ii)
**decreases

{Since the
total energy decreases, the radius of orbit also decreases.

ET = – GMmS / 2x

The total
energy decreases, that is, it becomes more negative. The value of (GMmS / 2x) should be greater so that the total
energy (with its negative sign) decreases. Since the value of (GMmS / 2x) increases, the radius of orbit, x,
decreases.}

**(iii)
**decreases

{The smaller
the radius of orbit, the lower the potential energy. Recall that the potential
energy is maximum (and equal to zero) at infinity.}

**(iv)
**increases

{E_{K} = GMm_{s} / 2x

From the
formula, as the radius x decreases, the E_{K} increases.}