Newtons law of universal gravitation:

$\overline{){\mathbf{F}}{\mathbf{=}}\frac{{\mathbf{GM}}_{\mathbf{1}}{\mathbf{M}}_{\mathbf{2}}}{{\mathbf{R}}^{\mathbf{2}}}}$

R = (a^{2} + x^{2})^{(1/2)}

$\mathit{F}\mathbf{=}\frac{\mathbf{G}\mathbf{M}\mathbf{m}}{\mathbf{(}{\mathbf{a}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}\mathbf{)}}$

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 (Figure 1).

Part A

What is the gravitational force that the sphere exerts on the ring-shaped object?

Express your answer in terms of the variables *a*, *M*, *m*, *R*, *x*, and appropriate constants.

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