Suppose a circular ring of uniform mass density and having mass and radius .
Now suppose a point mass (in rest) of mass initially at a distance of from its center lying on it's axis as shown below :
What will be the distance of the point mass from the center (measuring positive above while negative for below) after time have passed assuming that and only the forces of gravity of the ring and point masses are involved?
My work :
Acceleration of ring due to the gravity of point mass =
If is the velocity of the point mass and as it's gravitational potential energy, then Now, if the point mass have moved from a distance of from the center to a distance then from Work energy Theorem we have : Now as it is given that i.e. the point mass it at rest initially Now I just can't solve the integral. The integral can also be written as : Where and
Can someone solve this integral, or may have any other way of solving this?