Two satellite A and B, ratio of masses 3:1 are in circular orbits of radii r and 4 r. Then ratio of total mechanical energy of A to B is

To find the ratio of the total mechanical energy of satellites A and B, we first need to understand how mechanical energy in circular orbits is calculated. The total mechanical energy (E) of a satellite in orbit is given by the formula:

Mechanical Energy Formula

The total mechanical energy is the sum of kinetic energy (KE) and potential energy (PE). For a satellite in a circular orbit, the total mechanical energy can be expressed as:

• Potential Energy (PE): PE = -G(Mm)/r
• Kinetic Energy (KE): KE = (1/2)mv² = (GMm)/(2r)

Thus, the total mechanical energy (E) is:

E = KE + PE = (GMm)/(2r) - (GMm)/r = - (GMm)/(2r)

Calculating for Satellites A and B

Let the mass of satellite A be 3m and that of satellite B be m. The radii of their orbits are r for A and 4r for B.

Now, we can calculate the total mechanical energy for each satellite:

• For Satellite A:

E_A = - (G * (3m) * M) / (2r) = - (3GMm) / (2r)

• For Satellite B:

E_B = - (G * m * M) / (2 * 4r) = - (GMm) / (8r)

Finding the Ratio

Now, we can find the ratio of the total mechanical energy of A to B:

Ratio = E_A / E_B = [-(3GMm) / (2r)] / [-(GMm) / (8r)]

This simplifies to:

Ratio = (3/2) * (8/1) = 12

Final Result

The ratio of the total mechanical energy of satellite A to satellite B is 12:1.