How can I find the distance?

Mountain pull. A large mountain can slightly affect the direction of “down” as determined by a plumb line. Assume that we can model a mountain as a sphere of radius R = 2.00 km and density (mass per unit volume) 2.6 × 10^3 kg/m3. Assume also that we hang a 0.5 m plumb line at a distance of 3R from the sphere's center and such that the sphere pulls horizontally on the lower end. How far would the lower end move toward the sphere?

I came to the conclusion that it needs to be G*M/ r^2*g=tan (theta)
I cannot find a way to solve for distance.

force gravity= GMearth*m/reart^2

force mountain= GMmountain*m/(2e3)^2

arctanTheta= forcemountain/forceEarth
= check this
= MassMountain/massEarth*(radiusearth/2000)^2

now you have theta. notice the angle is small, use the small angle approximation (sinT=tanT)
distance/.5=sinTheta
distance= .3*tanTheta, tan theta above.

To solve for the distance the lower end moves toward the sphere, we need to consider the gravitational force between the sphere and the plumb line.

First, let's calculate the gravitational force using the equation F_grav = G * (m1 * m2) / r^2, where G is the gravitational constant, m1 and m2 are the masses involved, and r is the distance between the centers of the objects.

In this case, the plumb line has a very small mass compared to the mountain sphere, so we can consider the mass of the plumb line as negligible. Therefore, m1 in the equation above will be the mass of the sphere, and m2 will be the mass of the plumb line, which is negligible.

To find the gravitational force, we need to find the mass of the mountain sphere. The mass can be calculated using the equation mass = density * volume.

The volume of a sphere can be calculated using the formula V = (4/3) * pi * r^3, where r is the radius of the sphere.

Now, substitute the values into the equation to find the mass of the mountain sphere.

Once you have the mass of the sphere, you can calculate the gravitational force between the sphere and the plumb line using the gravitational force equation mentioned above.

Next, consider the force causing the plumb line to move toward the sphere. This force is the horizontal component of the gravitational force mentioned earlier. The horizontal component of a force can be calculated using the equation horizontal component = force * tan(theta), where theta is the angle at which the force is acting.

Now, you need to find the distance the lower end of the plumb line moves toward the sphere. This can be calculated using the equation for displacement: displacement = (1/2) * a * t^2, where t is the time taken for the motion to occur.

In this case, the acceleration, a, can be calculated using the equation acceleration = horizontal force / mass of the plumb line.

Using these steps, you can calculate the distance the lower end moves toward the sphere.