Question

An adult elephant has a mass of about 5.0 tons. An adult elephant shrew has a mass of about 50 grams. How far r from the center of the Earth should an elephant be placed so that its weight equals that of the elephant shrew on the surface of the Earth? The radius of the Earth is 6400 km. (1ton=103kg.)

Answer 1

Answer:

649529 km

Explanation:

Recall that the Universal Gravitational force between two masses $M_1$ and $M_2$ separated by a distance "d" is given by the equation: $F_g=G\frac{M_1\,M_2}{d^2}$ where G is the Universal Gravitational constant.

If we call the mass of the Earth $M_E$, the mass of the shrew m and the distance at the surface of the Earth R (for the radius of the Earth), then this equation becomes: $F_g=G\frac{M_E\,*\,m}{R^2}$

Now we want to find at which distance "D" the big elephant (of mass M) should be placed to experience an exactly equal gravitational force as the one we just estimated for the shrew, that is we want the following two expressions to be equal:

$G\frac{M_E\,*\,m}{R^2}=G\frac{M_E\,*\,M}{D^2}$

Notice that the factors G and $M_E$ that appear on both sides, can be cancelled out, and we are left with an equation where we can find the unknown distance "D" by solving for its square and then finding its square root:

$G\frac{M_E\,*\,m}{R^2}=G\frac{M_E\,*\,M}{D^2}\\\frac{m}{R^2}=\frac{M}{D^2}\\D^2=\frac{M*R^2}{m} \\D=\sqrt{\frac{M*R^2}{m} } \\D=R\sqrt{\frac{M}{m} }$

Now, before making the appropriate replacements for the calculation, let's make sure that we get the mass units all converted into kilograms for example, so they can cancel out in the quotient inside the root:

M = 5 tons = 5 * 103 kg

m = 50 grams = 0.05 kg

Therefore the quotient M/m is: 10300 and our equation for the distance "D" becomes: $D=R\sqrt{\frac{M}{m} }\\D=6400\,\sqrt{10300} \\D=649529\,km$