Consider a pair of planets for which the distance between them is decreased by a factor of 5. Show that the force between them becomes 25 times greater.
F=GMm/r^2
what happens if r is now 5r?
1/5
To show that the force between two planets becomes 25 times greater when their distance is reduced 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:
F = G * (m1 * m2) / r^2
Where G is the gravitational constant.
Let's consider two planets with masses m1 and m2, initially separated by a distance r. The force between them can be represented as F1:
F1 = G * (m1 * m2) / r^2
Now, when their distance is decreased by a factor of 5, the new distance between them becomes r/5. We want to find the new force between them, which we'll call F2:
F2 = G * (m1 * m2) / (r/5)^2
Simplifying the equation, we have:
F2 = G * (m1 * m2) / (r^2 / 25)
= (25 * G * (m1 * m2)) / r^2
From the equations for F1 and F2, we can observe that:
F2 = 25 * F1
This shows that the force between the planets becomes 25 times greater when their distance is decreased by a factor of 5.