consider a pair of planets that find that the distance between then is decreased by a factor of 5. Show that the force between them becomes 25 times as strong.
To show that the force between two planets becomes 25 times as strong when the distance between them is decreased by a factor of 5, we can use Newton's law of universal gravitation.
Newton's law of universal gravitation states that the force (F) between two objects is directly proportional to the product of their masses (m1 and m2) and inversely proportional to the square of the distance (r) between their centers of mass. The equation can be expressed as:
F = G * (m1 * m2) / r^2
Where:
- F is the force between the two objects,
- G is the gravitational constant,
- m1 and m2 are the masses of the two objects, and
- r is the distance between their centers of mass.
Let's assume the original distance between the planets is r0. Decreasing the distance by a factor of 5 means the new distance (r') is r0/5.
Now, let's compare the forces before (F0) and after (F') decreasing the distance between the planets.
F0 = G * (m1 * m2) / (r0^2)
F' = G * (m1 * m2) / ((r0/5)^2)
To simplify the comparison, let's divide F' by F0:
F' / F0 = (G * (m1 * m2) / ((r0/5)^2)) / (G * (m1 * m2) / (r0^2))
Now, cancel out similar terms:
F' / F0 = ((m1 * m2) / ((r0/5)^2)) * ((r0^2) / (m1 * m2))
The mass terms (m1 * m2) cancel out, and after simplification, we have:
F' / F0 = (r0^2 / (r0/5)^2)
Using the fact that (a/b)^2 = (a^2) / (b^2), we can rewrite the equation as:
F' / F0 = (r0^2 * 5^2) / (r0^2) = 25
Therefore, the force between the two planets becomes 25 times as strong when the distance between them is decreased by a factor of 5.