Physics GRAVITATION

Motion of a Satellite

Consider a satellite of mass `m` revolving in a circle around the earth. If the satellite is at a height `h` above the earth's surface, the radius of its orbit is `r = R + h` , where `R` is the radius of the earth. The gravitational force between `m` and `M` provides the necessary centripetal force for circular motion.

`text((a))` `text(Orbital Velocity)` `(v_0)`

The velocity of a satellite in its orbit is called orbital velocity.

Let `v_0` be the orbital velocity of the satellite, then

`(GMm)/r^2 = (mv_0^2)/r => v_0 = sqrt((GM)/r).................(1)`

or `v = sqrt((GM)/(R+h))` `[therefore r = R +h]`

`text(Important points :)`

Orbital velocity is independent of the mass of the orbiting body and is always along the tangent to the orbital.

Close to the surface of the earth, `r= R` as `h = 0` .

`v_0 = sqrt((GM)/R) = sqrt(gR) = sqrt(10 xx 6.4 xx 10^8) = approx 8 km//s`

`text((b))` `text(Time Period of a Satellite)`

The time taken by a satellite to complete one revolution is called the time period `(T)` of the satellite.

It is given by `T = (2 pi r)/v = 2 pi r sqrt(r/(GM))` or `T = (2 pi r sqrtr)/(sqrt(GM))`

or `T^2 = ((4 pi^2)/(GM))r^3...........................(2)`

`=> T^2 prop r^3`

`text((c))` `text(Angular Momentum of a Satellite ( L))`

In case of satellite motion, angular momentum will be given by

`L = mvr = mr sqrt((GM)/r)`

`L = (m^2 G M r)^(1/2)........................(3)`

In case of satellite motion, the net force on the satellite is always towards the center of the circular orbit. The torque of this force about the center of the orbit is zero. Hence, angular momentum of the satellite is conserved, i.e.,

`L =` constant

Energy of a Satellite

The P.E. of a satellite is

`U = mV = - (GMm)/r [since V = - (GM)/r]`

The kinetic energy of the satellite is

`k = 1/2 mv_0^2 = (GMm)/(2r)[since v_0 = sqrt((GM)/r)] `

Total mechanical energy of the satellite

`=-(GMm)/r+(GMm)/(2r) = - (GMm)/(2r)..........................(4)`

`text(Important points)`

We have,

`K/E = -1` i.e, `K = -E` Also, `U/E = 2 => U = 2E`

Total energy of a satellite in its orbit is negative. Negative energy means that the satellite is bound to the central body by an attractive force and energy must be supplied to remove it from the orbit to infinity.

`text(Binding energy of the satellite :)`

The energy required to remove the satellite from its orbit to infinity is called binding energy of the satellite, i.e.,

Binding energy `= - E = (GMm)/(2r)`

Weightlessness in a Satellite :

The radial acceleration of the satellite is given by

`a_r = F_r/m = (GMm)/(r^2 xx m) = (GM)/r^2`

For an astronaut inside the satellite, we have

`(GMm_a)/r^2 - N - m_aa_r =0`

where `m` , is mass of astronaut `a` , is radial acceleration of satellite and N is normal reaction on the astronaut

or `(GMm_a)/r^2 - N - (GMm_a)/r^2 =0`

`=>N= 0`

Hence, the astronaut feels weightlessness.