# 2.5E: Exercises for Section 6.5

• Gilbert Strang & Edwin “Jed” Herman
• OpenStax

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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

### Density Problems

In exercises 7 - 11, find the mass of the one-dimensional object.

7) A wire that is $$2$$ ft long (starting at $$x=0$$) and has a density function of $$ρ(x)=x^2+2x$$ lb/ft

8) A car antenna that is $$3$$ ft long (starting at $$x=0)$$ and has a density function of $$ρ(x)=3x+2$$ lb/ft

$$\frac{39}{2}$$

9) A metal rod that is $$8$$ in. long (starting at $$x=0$$) and has a density function of $$ρ(x)=e^{1/2x}$$ lb/in.

10) A pencil that is $$4$$ in. long (starting at $$x=2$$) and has a density function of $$ρ(x)=\dfrac{5}{x}$$ oz/in.

$$\ln(243)$$

11) A ruler that is $$12$$ in. long (starting at $$x=5$$) and has a density function of $$ρ(x)=\ln(x)+(1/2)x^2$$ oz/in.

In exercises 12 - 16, find the mass of the two-dimensional object that is centered at the origin.

12) An oversized hockey puck of radius $$2$$ in. with density function $$ρ(x)=x^3−2x+5$$

$$\frac{332π}{15}$$

13) A frisbee of radius $$6$$ in. with density function $$ρ(x)=e^{−x}$$

14) A plate of radius $$10$$ in. with density function $$ρ(x)=1+\cos(πx)$$

$$100π$$

15) A jar lid of radius $$3$$ in. with density function $$ρ(x)=\ln(x+1)$$

16) A disk of radius $$5$$ cm with density function $$ρ(x)=\sqrt{3x}$$

$$20π\sqrt{15}$$

### Spring Work Problems

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

18) 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

19) 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?

20) 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.

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

22) 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

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

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

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

25) 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

26) [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

27) [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?

28) [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

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

### Hydrostatic Force and Pressure

30) [T] A rectangular dam is $$40$$ ft high and $$60$$ ft wide. 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.

a. $$3,000,000$$ lb,
b. $$749,000$$ lb

### Pumping Work Problems

31) [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.

32) [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

33) [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.

34) 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.

$$W = \dfrac{AρH^2}{2}$$

35) 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.

36) 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.