consider a pair of planets that find the distance between them is decreased by a factoer of 5. Show that the force between them becomes 25 times as strong?
Use Newton's law of gravitation.
g=(m_1)(m_2)(G)/(d^2)
m_1 = mass of first body
m_2 = mass of second body
G = gravitational constant (value is unimportant for this particular problem)
d = distance between bodies.
look what happens when you use (1/5)d in place of d
F2 = F1 /(1/5)^2 = F1/(1/25) = 25 F1
Law of Gravitation:
F = GM1M2/r^2
Let the first distance = (5r)
then,
F1 = GM1M2/(5r)^2
or
F1= GM1M2/25r^2
The second distance is 1/5 of 5r = r
and
Let F2 = GM1M2/r^2 (with distance = r)
What is the ratio of F2 to F1, or
F2/F1 = ? (divide the expression for F2 by the expression for F1)
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 the inverse square law formula for gravitational force and apply the given information.
The inverse square law states that the gravitational force between two objects is inversely proportional to the square of the distance between their centers of mass.
Mathematically, the formula for gravitational force can be written as:
F = G * (m1 * m2) / r^2
Where:
F = gravitational force
G = gravitational constant (approximately 6.674 × 10^-11 Nm^2/kg^2)
m1, m2 = masses of the two objects
r = distance between the centers of mass of the two objects
Now, let's consider the initial distance between the planets (r1) and the final reduced distance (r2). According to the given information, r2 = r1/5.
We need to compare the forces of the two scenarios:
1. Initial Force (F1): F1 = G * (m1 * m2) / r1^2
2. Final Force (F2): F2 = G * (m1 * m2) / r2^2 = G * (m1 * m2) / (r1/5)^2
Simplifying the expression for F2:
F2 = G * (m1 * m2) / (r1/5)^2
= G * (m1 * m2) / (r1^2/25)
= G * (m1 * m2) * (25/r1^2)
Now, to compare F1 and F2, we can take the ratio of F2 to F1:
F2 / F1 = (G * (m1 * m2) * (25/r1^2)) / (G * (m1 * m2) / r1^2)
= 25
Thus, we see that the force between the two planets becomes 25 times as strong when the distance between them is decreased by a factor of 5.