A small sphere X of mass M is placed a distance d from a point mass. The gravitational force on the sphere X is 90 N. Sphere X is removed and a second sphere Y of mass 4M is placed a distance 3d from the same point mass. What is the gravitational force on sphere Y?
I did:
90 = GMm/d
Force on Y = G4Mm/3d
I thought it was 120 N but I was wrong.
I assume the point Mass is M.
force= 90*4/3^2=40 N
gravity follows an inverse-square relationship
... you didn't square the d
3 times the distance means 1/9 of the force
4 times the mass means 4 times the force
90 N * 4/9 = ?
Aha, that makes sense. Thank you both.
To find the gravitational force on sphere Y, we can start by considering the force on sphere X and then use the principles of gravitational force to determine the force on sphere Y.
Given that the gravitational force on sphere X is 90 N, we can use Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2
Where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between them.
In this case, the equation becomes:
90 = G * (M * m) / d^2
We can rearrange the equation to solve for m:
m = (90 * d^2) / (G * M)
Now, let's consider sphere Y. The mass of sphere Y is 4M, and its distance from the point mass is 3d. Using the same equation, we can find the gravitational force on sphere Y:
Force on Y = G * [(4M) * m] / (3d)^2
Substituting the expression we found for m:
Force on Y = G * [(4M) * (90 * d^2) / (G * M)] / (3d)^2
Force on Y = (4 * 90 * d^2) / (9 * d^2)
Simplifying:
Force on Y = 40 N
Therefore, the gravitational force on sphere Y is 40 N.
Well, it seems like you've got the right idea with the formula for gravitational force! Let's see if we can figure out what went wrong.
In your first equation, you correctly wrote the gravitational force on sphere X as 90 N, given by GMm/d. However, when you try to calculate the force on sphere Y, you wrote G4Mm/3d.
The issue with that is the mass of sphere Y should actually be squared since the force is a result of the interaction between two masses. So, the correct formula to use for the force on Y would be GM(4M)/(3d)^2.
That simplifies to GM^2/(9d^2), which means the gravitational force on sphere Y would be one-ninth of the force on sphere X. Therefore, the correct answer should be 10 N, not 120 N.
Remember, gravity can be a bit "force-ful" with its calculations! Keep practicing and you'll orbit around the right answer in no time!