Potential Energy

We all know that if we take (Earth + Block) as the system and we are interested in finding thce e potential energy of the(Block + Earth) then we can proceed ahead as follows: Case 1 if the mass of the block is very very small as compared to earth As we know that all conservative force fields of nature are gradient of scalar functions which we call as potential function.

Isolating the system so that no external force acts on the system. The gravitational force between the earth and the block is conservative and the earth is heavy so we can neglect its acceleration and it will still be nearly a very good inertial frame. Taking (Earth + Block) simplified our lives because in this frame the work done on earth will be zero. the force mg does work (-mgh) on the block if it ascends through height h and hence the potential energy increases by mgh and on reversal (i.e.) on descending the P.E. of the system decreases .Now since the earth remains fixed all the time means that it does not move due to the force exerted by mass of the block i can say as per my convenience that P.E. of the block is mgh where height h is from reference level and there i can consider the P.E. to be zero because only changes in them are significant or maybe non trivial.

NOW MY QUESTION ARISES : WHAT IF THE MASS INTRODUCED i.e.THE BLOCK'S MASS WAS ENOUGH TO PRODUCE ACCELERATION IN EARTH AND MY FRAME OF REFERENCE IS NOT EARTH THE SYSTEM REMAINS THE SAME (Block + Earth). NOW MY QUESTION IS THAT NOW IN SOME OTHER INERTIAL FRAME THE WORK IS BEING DONE BY INTERNAL CONSERVATIVE FORCE ON EARTH AND BLOCK AS THEY GET PULLED TOWARDS EACH OTHER AND THERE EXISTS SOME POTENTIAL ENERGY CORRESPONDING TO THESE CONSERVATIVE FORCES. BUT THIS TIME EARTH ALSO MOVES AND SOME POTENTIAL ENERGY ALSO EXISTS DUE TO THE WORK DONE ON IT.

SO HOW TO SOLVE THIS QUESTION A small block of superdense material of half the mass of earth. It is situated at a height h (much smaller than earth's radius) from where it falls on the earth's surface.Find its speed when its height is reduced to h/2.the mass of earth is 6 x 10^24 kg.

PLS HELP ME WITH THIS PROBLEM ASAP @Steven Chase @Josh Silverman

Note by Rishi Tiwari
2 weeks, 1 day ago

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If the two masses are similar, you must discard the assumption of constant acceleration and use the general Newtonian gravity equation instead. Also, I'm assuming you wish to neglect any effects of General Relativity that may arise due to the super-dense material. Suppose that the block position is \( x_b \) and the position of the Earth's center is \(x_E \). Assume \( x_b - x_E \geq R_E \), where \(R_E\) is the Earth radius.

\[\frac{G M m}{(x_b - x_E)^2} = -m \, \ddot{x_b} \\ \frac{G M m}{(x_b - x_E)^2} = M \, \ddot{x_E}\]

Initially, \(x_b - x_E = R_E + h\), where \(R_E\) is the Earth radius and \(h\) is the distance of the block from the Earth's surface. You can numerically integrate (or try to solve analytically) to see what the speed is when \(x_b - x_E = R_E + \frac{h}{2}\).

You could also equate the change in potential energy to the combined kinetic energies:

\[G M m \Big ( \frac{1}{R_E + h/2} - \frac{1}{R_E + h} \Big ) = \frac{1}{2} m \, \dot{x_b}^2 + \frac{1}{2} M \, \dot{x_E}^2 \]

You would have to make other physical arguments to determine what fraction of the converted potential energy goes to the Earth kinetic energy, and what fraction goes to the block kinetic energy.

Steven Chase - 2 weeks, 1 day ago

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@Steven Chase @Josh Silverman @Brilliant Physics @Rajdeep Dhingra

Rishi Tiwari - 2 weeks, 1 day ago

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