Question

A planet revolves around a star in a circular orbit. Prove that the potential energy of the planet is twice its total energy

Answer 1

HERE IS YOUR ANSWER MATE:

LET THE VELOCITY OF THE PLANET BE      v = √GM/r   (here M= mass of the star)

∴ KINETIC ENERGY (K.E) = 1/2mv²

                                         = 1/2m(√GM/r)²

                                         = GMm/2r

∴ POTENTIAL ENERGY ( P.E) = -GMm/r

WE KNOW THAT ,

MECHANICAL ENERGY (M.E) = (K.E)+(P.E)

                                                 = GMm/2r + (-GMm/r)

                                                 = GMm/2r - GMm/r

                                                 = -GMm/2r

                                                                                 

FROM HERE , WE CAME TO KNOW  

                      (M.E) = -(K.E)--------------------------(1)

              AND, (-K.E) = 1/2(P.E)

                      => (M.E) = 1/2(P.E)  [ using (1) ]

                      => (P.E) = 2(M.E)

HENCE PROVED.

HOPE IT 'LL HELP YOU FRIEND.

Answer 2

Answer:

Potential energy of the planet is twice it's total energy.

[#NOTE: P.E =Potential Energy, K.E = Kinetic Energy & M.E = Mechanical Energy]

Explanation:

Let the velocity of the planet be,

v = √GM/r (here, M = mass of the star)

therefore, K.E = 1/2 mv²

or, K.E = 1/2 m (√GM/r)²

or, K.E = GMm/2r

therefore, P.E = -GMm/r

We know that, M.E = K.E + P.E

i.e. M.E = GMm/2r + (-GMm/r)

or, M.E = GMm/2r - GMm/r

or, M.E = -GMm/2r

From the above equation we can conclude that,

M.E = (-K.E) ---------------(i)

and (-K.E) = 1/2 P.E

From (i) we get,

M.E = 1/2 P.E

therefore, P.E = 2 M.E.

[Hence, proved]

Hope this answer is helpful for you all !!