exercise:69f1d54288

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Suppose that a straight cylindrical hole is bored from the surface of the earth through the center and out the other side. An object of mass [math]m[/math] inside the hole and at a distance [math]r[/math] from the center of the earth is attracted to the center by a gravitational force equal in absolute value to [math]\frac{mgr}{R}[/math], where [math]g[/math] is constant and [math]R[/math] is the radius of the earth. Compute the work done by this force of gravity in terms of [math]m[/math], [math]g[/math], and [math]R[/math] as the object falls

• from the surface to the center of the earth,
• from the surface of the earth through the center to a point halfway between the center and surface on the other side,
• all the way through the hole from surface to surface.

[Hint: Let the [math]x[/math]-axis be the axis of the cylinder, and the origin the center of the earth. Define the gravitational force [math]F(x)[/math] acting on the object at [math]x[/math] so that: (i) its absolute value agrees with the above prescription, and (ii) its sign agrees with the convention given at the beginning of work.]