Question 1

What is the minimum energy required to launch a satellite of mass  m  from the surface of a planet of mass  M  and radius  R  in a circular orbit at an altitude of  2R ?

Accepted Answer

Given that,Mass of satellite  =m Mass of planet  =M Radius  =R Altitude  h=2R Now,The gravitational potential energy P.E=\dfrac{-Gm}{r} Potential energy at altitude  =\dfrac{GmM}{3R} Orbital velocity  {{v}_{0}}=\sqrt{\dfrac{2GmM}{R+h}} Now, the total energy is    {{E}_{f}}=\dfrac{1}{2}mv_{0}^{2}-\dfrac{GmM}{3R}     {{E}_{f}}=\dfrac{1}{2}\dfrac{GmM}{3R}-\dfrac{GMm}{3R}     {{E}_{f}}=\dfrac{GmM}{3R}\left[ \dfrac{1}{2}-1 \right]     {{E}_{f}}=\dfrac{-GmM}{6R}  Now,  {{E}_{i}}={{E}_{f}} Now, the minimum required energy    K.E=\dfrac{Gmm}{R}-\dfrac{GmM}{6R}     K.E=\dfrac{5GmM}{6R}  Hence, the minimum required energy is  \dfrac{5GmM}{6R}

Question 2

lf  \mathrm{W}_{1},\ \mathrm{W}_{2}  and  \mathrm{W}_{3}  represent the work done in moving a particle from A to  \mathrm{B}  along three different paths 1, 2 and 3 respectively (as shown) in the gravitational field of a point mass  \mathrm{m} , find the correct relation between  \mathrm{W}_{1},\ \mathrm{W}_{2}  and  \mathrm{W}_{3} .

Accepted Answer

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Gravitation

Physics
1848 Views

What is the minimum energy required to launch a satellite of mass $m$ from the surface of a planet of mass $M$ and radius $R$ in a circular orbit at an altitude of $2 R$ ?

lf $W_{1}, W_{2}$ and $W_{3}$ represent the work done in moving a particle from A to $B$ along three different paths 1, 2 and 3 respectively (as shown) in the gravitational field of a point mass $m$ , find the correct relation between $W_{1}, W_{2}$ and $W_{3}$ .

Gravitation

Physics 1848 Views

What is the minimum energy required to launch a satellite of mass m from the surface of a planet of mass M and radius R in a circular orbit at an altitude of 2R?

lf W1​, W2​ and W3​ represent the work done in moving a particle from A to B along three different paths 1, 2 and 3 respectively (as shown) in the gravitational field of a point mass m, find the correct relation between W1​, W2​ and W3​.

Physics
1848 Views

What is the minimum energy required to launch a satellite of mass $m$ from the surface of a planet of mass $M$ and radius $R$ in a circular orbit at an altitude of $2 R$ ?

lf $W_{1}, W_{2}$ and $W_{3}$ represent the work done in moving a particle from A to $B$ along three different paths 1, 2 and 3 respectively (as shown) in the gravitational field of a point mass $m$ , find the correct relation between $W_{1}, W_{2}$ and $W_{3}$ .