Object A experiences a gravitational force due to object B. How would that force change if the separation distance is halved, mA triples, and mB is reduced to one quarter of its original value?
g varies with the inverse square. For example if the distance is halved g is quadrupled.
To determine how the gravitational force between object A and object B changes, we can use the expression for the gravitational force:
F = G * (mA * mB) / r^2
Where:
F - Gravitational force
G - Universal gravitational constant (approximately 6.67430 × 10^-11 N m^2/kg^2)
mA - Mass of object A
mB - Mass of object B
r - Separation distance between object A and B
Let's break down each change and see how it affects the gravitational force.
1. If the separation distance is halved (r/2):
Plugging this into the formula, the gravitational force will become:
F' = G * (mA * mB) / (r/2)^2
= G * (mA * mB) / (r^2/4)
= 4 * (G * (mA * mB) / r^2)
= 4F
Therefore, halving the separation distance will increase the gravitational force by a factor of 4.
2. If object A's mass triples (3mA):
Plugging this into the formula, the gravitational force will become:
F'' = G * (3mA * mB) / r^2
= 3 * (G * (mA * mB) / r^2)
= 3F
Therefore, tripling object A's mass will increase the gravitational force by a factor of 3.
3. If object B's mass is reduced to one quarter (mB/4):
Plugging this into the formula, the gravitational force will become:
F''' = G * (mA * (mB/4)) / r^2
= (G * (mA * mB)) / (4r^2)
= (1/4) * (G * (mA * mB) / r^2)
= (1/4)F
Therefore, reducing object B's mass to one quarter will decrease the gravitational force by a factor of 1/4.
Combining all the changes, we get the final expression for the new gravitational force:
F_final = F' * F'' * F'''
= 4F * 3F * (1/4)F
= 12F * (1/4)F
= 3F * (1/4)F
= 3/4 * F^2
Hence, the new gravitational force will be reduced by a factor of 1/16 (3/4 * 3/4) compared to the original gravitational force.
To determine how a gravitational force changes under specific conditions, we need to use Newton's Law of Universal Gravitation. This law states that the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Let's denote the original force of gravity between object A and object B as F, the original separation distance as d, the original mass of object A as mA, and the original mass of object B as mB.
According to the problem, the given changes are:
1. The separation distance is halved, so the new distance (d') is d/2.
2. The mass of object A triples, so the new mass of A (mA') is 3 times the original mass (3mA).
3. The mass of object B is reduced to one quarter of its original value, so the new mass of B (mB') is one-fourth the original mass (1/4 * mB).
To determine the new force of gravity (F'), we need to consider each change independently and apply the updated values to Newton's Law of Universal Gravitation.
Change 1: Halving the separation distance
According to Newton's Law of Universal Gravitation, the force of gravity is inversely proportional to the square of the distance, so reducing the separation distance by half will cause the force of gravity to increase by a factor of (2^2) = 4.
Therefore, the new force after halving the separation distance is 4F.
Change 2: Tripling the mass of object A
Increasing the mass of object A will cause the force of gravity to increase proportionally. Therefore, the new force after tripling the mass of object A is (3mA) times the original force, which is 3F.
Change 3: Reducing the mass of object B to one-fourth its original value
Reducing the mass of object B will cause the force of gravity to decrease proportionally. Therefore, the new force after reducing the mass of object B is (1/4 * mB) times the original force, which is (1/4 * F).
Combining all three changes, the total effect on the force of gravity is as follows:
F' = (4F) * (3F) * (1/4 * F)
F' = (4 * 3 * 1/4) * (F * F * F)
F' = 3F * F * F
F' = 27F
Hence, the new force of gravity (F') is 27 times greater than the original force (F) under the given changes.