CALC An object in the shape of a thin ring has radius a and mass M. A uniform sphere with mass m and radius R is placed with its center at a distance x to the right of the center of the ring, along a line through the center of the ring, and perpendicular to its plane (see Fig. E13.35). What is the gravitational force that the sphere exerts on the ring-shaped object? Show that your result reduces to the expected result when x is much larger than a.

Solution to 83P Step 1 of 4 Introduction We will consider a small infinitesimal mass dm on the ring and will compute the gravitational force by the small sphere on dm. Then using integration we will find the total force of gravitation. Let y be the line joining dm to the small sphere. Y makes angle with the radius. Force could be resolved into two components. which will cancel out in all direction. which will add up. Step 2 of 4 Step 3 of 4 We know that the force on dm is given by, Step 4 of 4 R<