Question
Can someone please explain the following situation? Calculations orsome type of work would be appreciated.
It perhaps doesn't come as a surprise that our old gravitypotential energy expression, PE = mgh, is only good when you'realso very close to the surface of the earth. The more generalexpression for the potential energy due to gravity is PE =-Gm1m2/r (where r is an always-positivedistance between the centers of the two masses).
Your job in this essay is to work a quick sample case to show thismore general PE expression makes sense. Demonstrate how this newexpression still has your gravitational potential energy decreasingas your falling object (say, a 5000kg asteroid) gets closer to theearth. If you're losing PE, then the asteroid must be gaining KE(since there are no other forces at play that can do work on theasteroid). The asteroid must, then, speed up as it approachesearth, which makes sense.
Explanation / Answer
The element of the first equation, PE = mgh, that causes theproblem is g, is acceleration due to gravity. This value changeswith distance. Near the Earth's surface, the value changes onlyslightly, so mgh becomes a good estimate. We can solve for theactual equation that g represents. Fg =GMm/r2 mag=GMm/r2 ag= g = GM/r2 Entering this into our estimate equation, PE = mgh =GMmh/r2 = GMm/r (h =r) This shows that our two equations are equivalent near theearth's surface (ignoring the arbitrary sign in the secondequation, whose importance is demonstrated below). As the asteroid falls to earth, h and r decrease over time.For PE = mgh, this results in a smaller number, and less PE meansmore KE, matching observations. For PE = -GMm/r, as r decreases,the value of GMm/r becomes larger. Because of this, thenegative sign is applied and the value becomes more negative, alesser number, matching our expectations. This is the reason forthe neccessary negative sign. Entering this into our estimate equation, PE = mgh =GMmh/r2 = GMm/r (h =r) This shows that our two equations are equivalent near theearth's surface (ignoring the arbitrary sign in the secondequation, whose importance is demonstrated below). As the asteroid falls to earth, h and r decrease over time.For PE = mgh, this results in a smaller number, and less PE meansmore KE, matching observations. For PE = -GMm/r, as r decreases,the value of GMm/r becomes larger. Because of this, thenegative sign is applied and the value becomes more negative, alesser number, matching our expectations. This is the reason forthe neccessary negative sign.