Question

Consider Earth to be stationary, and the moon as orbiting Earth in a circle of radius R. If the masses of Earth and the Moon are ME and Mm, respectively, which of the following best represents the total mechanical energy of the Earth-Moon system?a.
`(GM_EM_M)/(2R)`

b.
`(GM_EM_M)/(R)`
c.
`-(GM_EM_M)/(2R)`
d.
`-(GM_EM_M)/(R)`
e.
`-(2GM_EM_M)/(R)`

Answer 1

The total mechanical energy of the Earth-Moon system is represented by the formula -GM_E*M_m/2R.

Step-by-step explanation:

The total mechanical energy of a two-body system, such as the Earth-Moon system, is the sum of the kinetic and potential energies of the system. To calculate this, we use the equation for gravitational potential energy, U = -(G * ME * Mm)/r, and the kinetic energy, K = U/2. This comes from the fact that for circular orbits, the kinetic energy is half the magnitude of the gravitational potential energy but positive in value, so when combined, the total mechanical energy is E = K + U = -U/2. Therefore, the total mechanical energy of the Earth-Moon system is represented by -GMEMm/2R.

Answer 2

option C

Step-by-step explanation:

given,

mass of earth = Me

mass of the moon = Mm

radius = R

Total mechanical energy = ?

Kinetic energy of moon =

`KE = (1)/(2)M_m v^2`

velocity of the moon is equal to

`v = \sqrt{(GM_e)/(R)}`

total energy =
`-(1)/(2)M_m v^2`

now, putting value of in the above equation

`TE = -(1)/(2)M_m (\sqrt{(GM_e)/(R)})^2`

`TE = -(1)/(2)M_m(GM_e)/(R)`

`TE = -(1)/(2)(GM_mM_e)/(R)`

`TE = -(GM_mM_e)/(2R)`

hence, the correct answer is option C