Two spheres of mass

M1 = 810 kg and M2 = 390 kg are placed
4.80 m apart. A particle of mass m = 12.0 kg is now placed midway between the two spheres.

(a) What is the net gravitational force on the particle due to the two spheres? N toward the sphere of M1 or M2

(b) At what position between the two spheres should the particle be placed so that the net gravitational force on the particle is zero? m from the sphere of mass M1

man idek physics is so hard ugh

net force will be G(810-390)(12)/2.4^2 and pointed toward the larger mass.

f1=f2
G810(12)/x^2=G390*12/(4.8-x)^2

x^2(390)=(4.8-x)^2*810
solve for x.

(a) Well, if I had to guess, I'd say the net gravitational force on the particle is going to be pretty attracted to the sphere with a mass of 810 kg. After all, it's got more mass to throw around and create some gravitational pull. So, my professional clown opinion is that the net gravitational force is pointing toward the sphere with a mass of M1.

(b) Ah, the elusive zero net gravitational force point. If you want to cancel out that gravitational pull from the two spheres, you'll have to do a little balancing act. So, take your particle of mass 12.0 kg and place it at a distance of m (you didn't give me an exact number, so I'll leave that up to you) from the sphere with mass M1. Find that sweet spot where the gravitational forces from both spheres cancel each other out, and voila! You've achieved the elusive zero net gravitational force.

Just be careful not to slip and fall into a black hole while you're searching for that spot. Safety first, my friend!

To calculate the net gravitational force on the particle due to the two spheres and determine the position where the net force is zero, we need to apply the law of gravitation.

The law of 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 their centers.

Let's break down the steps to find the answers to the provided questions:

(a) Calculate the gravitational force on the particle due to each sphere individually.
1. Calculate the gravitational force between the particle and sphere M1 using the formula:
F1 = (G * M1 * m) / d1^2
- G is the gravitational constant (approximately 6.67 × 10^-11 N.m^2/kg^2).
- M1 is the mass of sphere M1 (810 kg).
- m is the mass of the particle (12.0 kg).
- d1 is the distance between the particle and sphere M1 (half the distance between the two spheres, 4.80 m / 2 = 2.40 m).
2. Calculate the gravitational force between the particle and sphere M2 using the same formula:
F2 = (G * M2 * m) / d2^2
- M2 is the mass of sphere M2 (390 kg).
- d2 is the distance between the particle and sphere M2 (half the distance between the two spheres, 4.80 m / 2 = 2.40 m).

(b) Determine the position between the two spheres where the net gravitational force on the particle is zero.
1. The net gravitational force is zero when the magnitudes of the forces from both spheres are equal.
|F1| = |F2|
2. Use these equations to solve for the position from the sphere of mass M1:
(G * M1 * m) / d1^2 = (G * M2 * m) / d2^2

Now let's plug in the values and calculate the answers:

(a) Calculation:
F1 = (6.67 × 10^-11 N.m^2/kg^2 * 810 kg * 12.0 kg) / (2.40 m)^2
F2 = (6.67 × 10^-11 N.m^2/kg^2 * 390 kg * 12.0 kg) / (2.40 m)^2

The net gravitational force can be found by finding the difference between these forces (as their directions will be opposite). If F1 > F2, the net force is directed toward the sphere of mass M1. If F2 > F1, the net force is directed toward the sphere of mass M2.

(b) Calculation:
(6.67 × 10^-11 N.m^2/kg^2 * 810 kg * 12.0 kg) / (d1)^2 = (6.67 × 10^-11 N.m^2/kg^2 * 390 kg * 12.0 kg) / (d2)^2

Solving this equation will give you the value of d1, which represents the position from the sphere of mass M1 where the net gravitational force on the particle is zero.