The amount of energy required to life the elephant over his head was 98 KJ In cl

Question

The amount of energy required to life the elephant over his head was 98 KJ

In class we calculated the work, i.e., the energy necessary for me to lift an elephant over my head. Now calculate the energy required to put an elephant into circular, low Earth orbit at altitude h above the Earth's surface. That is, calculate the difference between the total energy of the elephant in orbit and its total energy standing on the surface of the rotating Earth. Assume the elephant starts standing on the equator, and treat the Earth as a sphere.

Explanation / Answer

Hi,

In this case we could consider only the difference in the potential gravitational energy (as I think it was done when you calculated the energy required to lift the elephant). However, as they are mentioning the fact that the elephant will be rotating with the Earth both in the surface and in the orbit, then we should also consider the change in kinetic energy.

The energy of the elephant at the surface of the Earth is:

E1 = mgh1 + (1/2)mv12  

The energy of the elephant at the orbit is:

E2 = mgh2 + (1/2)mv22

If we put the reference for the height at the center of the Earth (considering it a sphere) and besides, we consider that the Earth radius has a value of 6370 km, we have the following:

h1 = R = 6370 km = 6.37*106 m

The maximum distance of an object in the low Earth orbit is about 2000 km, therefore:

h2 = R + 2000 km = 8370 km = 8.37*106 m

Moreover, as we treated the Earth as a sphere we can calculate its angular speed and with that we can find the velocities v1 and v2:

w = 2/(24 h) = 2/(86400 s) = 7.27*10-5 rad/s

v2 = h2w = (7.27*10-5 rad/s)(8.37*106 m) = 609 m/s

v1 = h1w = (7.27*10-5 rad/s)(6.37*106 m) = 463 m/s

Now, we only have to find the difference in energies:

E1 = (5000 kg)(9.8 m/s2 )(6.37*106 m) + (1/2)(5000 kg)(463 m/s)2 = 3.13*1011 J

E2 = (5000 kg)(9.8 m/s2 )(8.37*106 m) + (1/2)(5000 kg)(609 m/s)2 = 4.11*1011 J

E = E2 - E1 = 9.8*1010 J = 9.8*107 kJ

Note: The energy found is the maximum difference that can be gotten, as the value used for the low Earth orbit was the maximum one. Additionaly, we can see that the effect of the potential gravitational energy is much bigger than the kinetic one (being this one about a 0.2 % of the potential energy). Finally, the energy found is one million times bigger than the energy needed to lift the elephant.

I hope it helps.

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