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A shock absorber is compressed 1 in. by a weight of 1 t. What is the spring constant?

A force of $F=20x-{x}^{3}$ N stretches a nonlinear spring by $x$ meters. What work is required to stretch the spring from $x=0$ to $x=2$ m?

Find the work done by winding up a hanging cable of length $100$ ft and weight-density $5$ lb/ft.

For the cable in the preceding exercise, how much work is done to lift the cable $50$ ft?

For the cable in the preceding exercise, how much additional work is done by hanging a $200$ lb weight at the end of the cable?

[T] A pyramid of height $500$ ft has a square base $800$ ft by $800$ ft. Find the area $A$ at height $h.$ If the rock used to build the pyramid weighs approximately $w=100\,{\text{lb/ft}}^{3},$ how much work did it take to lift all the rock?

[T] For the pyramid in the preceding exercise, assume there were $1000$ workers each working $10$ hours a day, $5$ days a week, $50$ weeks a year. If the workers, on average, lifted 10 100 lb rocks $2$ ft/hr, how long did it take to build the pyramid?

[T] The force of gravity on a mass $m$ is $F=-((GMm)/{x}^{2})$ newtons. For a rocket of mass $m=1000\,\text{kg},$ compute the work to lift the rocket from $x=6400$ to $x=6500$ km. State your answers with three significant figures. (Note: $G=6.67\,\times \,{10}^{-11}\,{\text{N m}}^{2}/{\text{kg}}^{2}$ and $M=6\,\times \,{10}^{24}\,\text{kg}.)$

[T] For the rocket in the preceding exercise, find the work to lift the rocket from $x=6400$ to $x=\infty .$

[T] A rectangular dam is $40$ ft high and $60$ ft wide. Assume the weight density of water is 62.5 lbs/ft³. Compute the total force $F$ on the dam when

a. the surface of the water is at the top of the dam and b. the surface of the water is halfway down the dam.

[T] Find the work required to pump all the water out of a cylinder that has a circular base of radius $5$ ft and height $200$ ft. Use the fact that the density of water is $62$ lb/ft³.

[T] Find the work required to pump all the water out of the cylinder in the preceding exercise if the cylinder is only half full.

[T] How much work is required to pump out a swimming pool if the area of the base is $800$ ft², the water is $4$ ft deep, and the top is $1$ ft above the water level? Assume that the density of water is $62$ lb/ft³.

A cylinder of depth $H$ and cross-sectional area $A$ stands full of water at density $\rho .$ Compute the work to pump all the water to the top.

For the cylinder in the preceding exercise, compute the work to pump all the water to the top if the cylinder is only half full.

A cone-shaped tank has a cross-sectional area that increases with its depth: $A=(\pi {r}^{2}{h}^{2})/{H}^{3}.$ Show that the work to empty it is half the work for a cylinder with the same height and base.

What are the typical units of work?

If a man has a mass of 80 kg on Earth, will his mass on the moon be bigger, smaller, or the same?

If a woman weighs 130 lb on Earth, will her weight on the moon be bigger, smaller, or the same?

Fill in the blanks:

Some integrals in this section are set up by multiplying a variable _____ by a constant distance; others are set up by multiplying a constant force by a variable _____.