How would the force of gravity between two masses be affected if the separation distance between them were the following? (Assume F2 represents the new force and F1 represents the original force.)

(a) triple the original distance
F2 / F1 =
(b) one-fourth the original distance
F2 / F1 =

(a) Well, if the separation distance between the two masses is tripled, the force of gravity would definitely feel a bit lonely. It would become a third wheel, you know? F2 / F1 = Making you feel like you're missing some gravity action!

(b) Ah, if the separation distance is reduced to one-fourth the original distance, things would get pretty cozy for gravity. It'd be like a clingy relationship. F2 / F1 = Isn't it cute when gravity just can't stay away?

(a) If the separation distance between two masses is tripled, we can calculate the ratio of the new force (F2) to the original force (F1) using the formula for gravitational force:

F = (G * m1 * m2) / r^2

where G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the separation distance.

Let's assume that the original force is F1 when the separation distance is r and the new force is F2 when the separation distance is 3r.

F1 = (G * m1 * m2) / r^2

F2 = (G * m1 * m2) / (3r)^2

To calculate the ratio F2 / F1, we can substitute the expressions for F2 and F1:

F2 / F1 = ((G * m1 * m2) / (3r)^2) / ((G * m1 * m2) / r^2)

Simplifying the expression by canceling out like terms:

F2 / F1 = (r^2 / (9r^2)

F2 / F1 = 1 / 9

Therefore, when the separation distance is tripled, the force of gravity between the two masses is reduced to one-ninth (1/9) of the original force.

(b) If the separation distance between two masses is one-fourth of the original distance, we can use the same formula to calculate the ratio of the new force (F2) to the original force (F1):

Let's assume that the original force is F1 when the separation distance is r and the new force is F2 when the separation distance is (1/4)r.

F1 = (G * m1 * m2) / r^2

F2 = (G * m1 * m2) / ((1/4)r)^2

To calculate the ratio F2 / F1, we substitute the expressions for F2 and F1:

F2 / F1 = ((G * m1 * m2) / ((1/4)r)^2) / ((G * m1 * m2) / r^2)

Simplifying the expression by canceling out like terms:

F2 / F1 = ((1/16)r^2) / r^2

F2 / F1 = 1 / 16

Therefore, when the separation distance is one-fourth of the original distance, the force of gravity between the two masses is reduced to one-sixteenth (1/16) of the original force.

To determine how the force of gravity between two masses would be affected if the separation distance between them is changed, we can use the formula for gravitational force:

F = (G * m1 * m2) / r^2

where F represents the force of gravity between two masses (m1 and m2), G is the gravitational constant, and r is the separation distance between the centers of the two masses.

(a) If the separation distance is tripled, we can substitute 3r for r in the equation and simplify:

F2 = (G * m1 * m2) / (3r)^2
= (G * m1 * m2) / 9r^2

To find the ratio F2 / F1, we divide F2 by F1:

F2 / F1 = [(G * m1 * m2) / 9r^2] / [(G * m1 * m2) / r^2]
= (G * m1 * m2) / 9r^2 * r^2 / (G * m1 * m2)
= 1 / 9

Therefore, F2 / F1 = 1 / 9. The force of gravity is one-ninth (1/9) of the original force when the separation distance is tripled.

(b) If the separation distance is one-fourth the original distance, we can substitute (1/4)r for r in the equation and simplify:

F2 = (G * m1 * m2) / ((1/4)r)^2
= (G * m1 * m2) / (1/16)r^2
= 16(G * m1 * m2) / r^2

To find the ratio F2 / F1, we divide F2 by F1:

F2 / F1 = [16(G * m1 * m2) / r^2] / [(G * m1 * m2) / r^2]
= 16

Therefore, F2 / F1 = 16. The force of gravity is sixteen times (16) the original force when the separation distance is one-fourth the original distance.

((1/3)^2 = 1/9

4^2 = 16