Question

An artificial satellite circles the Earth in a circular orbit at a location where the acceleration due to gravity is 6.44 m/s2. Determine the orbital period of the satellite.

Answer 1

Step-by-step explanation:

The artificial satellite experiences a centripetal force
`F_c` as it moves around the earth and it is defined as

`F_c = m(v^2)/(r) = m\left((2\pi r)/(T)\right)^2\left((1)/(r)\right) = (4\pi^2mr)/(T^2)`

where m is the mass of the satellite, r is its orbital radius and T is its orbital period. But we need to find the radius first.

Recall that the satellite is orbiting at a height where its acceleration due to gravity is 6.44 m/s^2. Since we know that the weight mg of the satellite is equal to the gravitational force
`F_G` between the earth and the satellite, we can write

`mg = F_G = G(mM)/(r^2)`

`\Rightarrow g = G(M)/(r^2)`

where M is the mass of the earth (=
`5.972×10^(24)\:\text{kg}`) and G is the universal gravitational constant (=
`6.674×10^(-11)\:\text{N-m}^2\text{/kg}`). Plugging in the values, we find that the radius of the satellite's orbit is

`r = \sqrt{(GM)/(g)} = \sqrt{\frac{(6.674×10^(-11)\:\text{N-m}^2\text{/kg})(5.972×10^(24)\:\text{kg})}{6.44\:\text{m/s}^2}}`

`\:\:\:\:\:= 7.87×10^6\:\text{m}`

Now that we have the value for the radius, we can now calculate the orbital period T. Recall that the centripetal force is equal to the weight of the satellite at its orbital radius. Therefore,

`F_c = mg \Rightarrow (4\pi^2mr)/(T^2) = mg`

or

`4\pi^2r = gT^2`

Solving for T, we get

`T^2 = (4\pi^2r)/(g) \Rightarrow T = \sqrt{(4\pi^2r)/(g)}`

We can further simplify the above expression into

`T = 2\pi\sqrt{(r)/(g)}`

Plugging in the values for r and g, we get

`T = 2\pi\sqrt{\frac{(7.87×10^6\:\text{m})}{(6.44\:\text{m/s}^2)}}`

`\:\:\:\:\:= 6945\:\text{s} = 1.93\:\text{hrs}`