2.5E: Exercises for Section 6.5

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?

Answer

$W = 150$ 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.

Answer

$W = 200$ 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?

Answer

$W = 1$ 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

Answer

$\frac{39}{2}$

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.

Answer

$\ln(243)$

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 $ρ(x)=x^3−2x+5$

Answer

$\frac{332π}{15}$

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

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

Answer

$100π$

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 $ρ(x)=\sqrt{3x}$

Answer

$20π\sqrt{15}$

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?

Answer

$W = 6$ 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?

Answer

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?

Answer

$W = 36$ 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?

Answer

$W = 18,750$ 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?

Answer

$W= \frac{32}{3}×10^9$ 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.)

Answer

$W = 8.65×10^5$ 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.

Answer

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.

Answer

$W = 23.25π$ 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.

Answer

$W = \dfrac{AρH^2}{2}$

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.

Answer

Answers may vary.

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.