The force of gravitational attraction (Fg) between two bodies is directly proportional to the product of the two masses (m1, m2) and inversely proportional to the square of the distance (d) separating the two masses is given by the formula below. What happens to the gravitational force ,Fg, if the distance, d, is quartered ? Everything else in the formula is constant.
Note: α is the centripetal acceleration and T is the period
gm1m2/d2 = Fg
If the distance, d, is quartered, it means that the new distance (d') is one-fourth of the original distance (d).
Using the formula for gravitational force (Fg), which states that Fg is inversely proportional to the square of the distance (d), we can determine the effect on the gravitational force.
If d' = d/4, then (d')^2 = (d/4)^2 = (d^2)/16.
Substituting (d')^2 into the formula for gravitational force (Fg), we get:
Fg' = gm1m2/(d')^2 = gm1m2/((d^2)/16)
Now, let's simplify this expression:
Fg' = (gm1m2)/(d^2/16) = (gm1m2) * (16/d^2)
Since everything else in the formula is constant, we can say that the gravitational force Fg' is directly proportional to 16/d^2.
If the distance, d, is quartered (d' = d/4), the new gravitational force, Fg', will increase by a factor of (16/d^2). In other words, the gravitational force will increase by a factor of 16 times the inverse of the square of the distance.
Therefore, if the distance is quartered, the gravitational force will increase.
If the distance, d, is quartered, it means that it is divided by 4. To understand what happens to the gravitational force, Fg, we can use the formula you provided:
Fg = gm1m2/d^2
Let's substitute d/4 for d in the formula:
Fg = gm1m2/(d/4)^2
Simplifying further:
Fg = gm1m2/(d^2/16)
Fg = 16gm1m2/d^2
Comparing this with the original formula, we can see that the gravitational force, Fg, increases by a factor of 16 when the distance, d, is quartered. Therefore, the gravitational force becomes 16 times stronger.