Compute the work in foot-pounds done by the force of gravity when a [math]50[/math]-pound rock falls [math]200[/math] feet off a vertical cliff.

Compute the work in foot-pounds done against the force of gravity in raising a [math]10[/math]-pound weight vertically [math]6[/math] feet from the ground.

A car on a horizontal track is attached to a fixed point by a spring, as shown in Figure. The spring has been stretched [math]2[/math] feet beyond its rest position, and the car is held there by a force of [math]10[/math] pounds. If the car is released, how many foot-pounds of work are done by the restoring force of the spring moving the car [math]2[/math] feet back to the rest position?

An electron is attracted to a nucleus by a force which is inversely proportional to the square of the distance [math]r[/math] between them; i.e., [math]\frac{k}{r^2}[/math]. If the nucleus is fixed, compute the work done by the attractive force in moving the electron from [math]r=2a[/math] to [math]r=a[/math].

A container holding water is raised vertically a distance of [math]10[/math] feet at the constant rate of [math]10[/math] feet per minute. Simultaneously water is leaking from the container at the constant rate of 15 pounds per minute. If the empty container weighs [math]1[/math] pound and if it holds [math]15[/math] pounds of water at the beginning of the motion, find the work done against the force of gravity.

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 \secref{8.5}.]

Consider a cylinder and piston as shown in Figure. The inner chamber, which contains gas, has a radius [math]a[/math] and lenght [math]b[/math]. According to the simplest gas law, the expansive force of the gas on the piston is inversely proportional to the volume [math]v[/math] of gas; i.e., [math]F = \frac{k}{v}[/math] for some constant [math]k[/math]. Compute the work done against this force in compressing the gas to half its initial volume by pushing in the piston.

A rocket of mass [math]m[/math] is on its way from the earth to the moon along a straight line joining their centers. Two gravitational forces act simultaneously on the rocket and in opposite directions. One is the gravitational pull toward earth, equal in absolute value to [math]\frac{GM_1m}{{r_1}^2}[/math], where [math]G[/math] is the universal gravitational constant, [math]M_1[/math] the mass of the earth, and [math]r_1[/math] the distance between the rocket and the center of the earth. The other is the analogous gravitational attraction toward the moon, equal in absolute value to [math]\frac{GM_2m}{{r_2}^2}[/math], where [math]M_2[/math] is the mass of the moon and [math]r_2[/math] is the distance between the rocket and the center of the moon. Denote the radii of the earth of the earth and moon by [math]a[/math] and [math]b[/math], respectively, and let [math]d[/math] be the distance between their centers.

• Take the path of the rocket for the [math]x[/math]-axis with the centers of earth and moon at [math]0[/math] and [math]d[/math], respectively, and compute [math]F(x)[/math], the resultant force acting on the rocket at [math]x[/math].
• Set up the definite integral for the work done against the force [math]F[/math] as the rocket moves from the surface of the earth to the surface of the moon.