if the masses remain the same ,but the separation is decreased to one-half of the original distance . what happens to the gravitation force?
f = GMm/r^2
if you replace r by r/2, you have
GMm/(r/2)^2 = 4GMm/r^2 = 4f
so, the force increases 4-fold
If r changes by a factor of k, the force changes by a factor of 1/k^2
F =G•m1•m2/R²
the gravitational constant
G =6.67•10⁻¹¹ N•m²/kg²,
R1=r/2 => F1=4F
To determine what happens to the gravitational force when the separation distance between two masses is decreased to one-half of the original distance while the masses remain the same, we can use Newton's law of universal gravitation.
Newton's law of universal gravitation states that the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Mathematically, the equation for gravitational force can be written as:
F = G * (m1 * m2) / r^2
Where:
- F is the gravitational force between the masses
- G is the gravitational constant (approximately 6.674 × 10^(-11) N*m^2/kg^2)
- m1 and m2 are the masses of the two objects
- r is the separation distance between the objects
In this case, if the masses remain the same but the separation distance is decreased to one-half of the original distance, we can denote the original separation distance as r1 and the new separation distance as r2. Therefore, r2 = 1/2 * r1.
We can now compare the gravitational forces by plugging these values into the equation:
F1 = G * (m1 * m2) / r1^2
F2 = G * (m1 * m2) / r2^2 (r2 = 1/2 * r1)
To find out what happens to the gravitational force, let's calculate the ratio of these two forces:
F2 / F1 = (G * (m1 * m2) / r2^2) / (G * (m1 * m2) / r1^2)
By cancelling out the common terms, we get:
F2 / F1 = r1^2 / r2^2
Substituting r2 = 1/2 * r1 into the equation:
F2 / F1 = (r1^2) / ((1/2 * r1)^2)
Simplifying further:
F2 / F1 = (r1^2) / ((1/4)*r1^2)
F2 / F1 = 4
Therefore, the ratio of the gravitational forces when the separation distance is decreased to one-half of the original distance is 4. This means that the gravitational force increases by a factor of 4 when the separation distance is decreased to one-half, given that the masses remain the same.