
# 6.5E: Exercises on Work

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## Basic Work Problems

For exercises 1 - 6, find the work done.

1) Find the work done when a constant force $$F=12$$ lb moves a chair from $$x=0.9$$ to $$x=1.1$$ ft.

2) How much work is done when a person lifts a $$50$$ lb box of comics onto a truck that is $$3$$ ft off the ground?

$$W = 150$$ ft-lb

3) What is the work done lifting a $$20$$ kg child from the floor to a height of $$2$$ m? (Note that $$1$$ kg equates to $$9.8$$ N)

4) Find the work done when you push a box along the floor $$2$$ m, when you apply a constant force of $$F=100$$ N.

$$W = 200$$ J

5) Compute the work done for a force $$F=\dfrac{12}{x^2}$$ N from $$x=1$$ to $$x=2$$ m.

6) What is the work done moving a particle from $$x=0$$ to $$x=1$$ m if the force acting on it is $$F=3x^2$$ N?

$$W = 1$$ J

## Spring Work Problems

7) A $$12$$-in. spring is stretched to $$15$$ in. by a force of $$75$$ lb. What is the spring constant?

8) A spring has a natural length of $$10$$ cm. It takes $$2$$ J to stretch the spring to $$15$$ cm. How much work would it take to stretch the spring from $$15$$ cm to $$20$$ cm?

$$W = 6$$ J

9) A $$1$$-m spring requires $$10$$ J to stretch the spring to $$1.1$$ m. How much work would it take to stretch the spring from $$1$$ m to $$1.2$$ m?

10) A spring requires $$5$$ J to stretch the spring from $$8$$ cm to $$12$$ cm, and an additional $$4$$ J to stretch the spring from $$12$$ cm to $$14$$ cm. What is the natural length of the spring?

The natural length is$$5$$ cm.

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

12) A force of $$F=\left(20x−x^3\right)$$ N stretches a nonlinear spring by $$x$$ meters. What work is required to stretch the spring from $$x=0$$ to $$x=2$$ m?

$$W = 36$$ J

## Cable and Chain Work Problems

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

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

$$W = 18,750$$ ft-lb

15) 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?

## Pyramid & Satellite/Rocket Work Problems

16) [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?

$$W= \frac{32}{3}×10^9$$ ft-lb

17) [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 each of the workers, on average, lifted ten 100-lb rocks $$2$$ft/hr, how long did it take to build the pyramid?

18) [T] The force of gravity on a mass $$m$$ is $$F=−((GMm)/x^2)$$ newtons. For a rocket of mass $$m=1000$$ kg, compute the work to lift the rocket from $$x=6400$$ to $$x=6500$$ km. (Note: $$G=6×10^{−17}\,\text{N m}^2/\text{kg}^2$$ and $$M=6×10^{24}$$ kg.)

$$W = 8.65×10^5$$ J

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

## Pumping Work Problems

20) [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.

21) [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.

$$W = 23.25π$$ million ft-lb

22) [T] How much work is required to pump out a swimming pool if the area of the base is $$800 \, \text{ft}^2$$, 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.

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

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