Physical Applications

For the following exercises (1-3), find the work done.

1. 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?
2. Find the work done when you push a box along the floor $2$ m, when you apply a constant force of $F=100\text{N}.$
3. What is the work done moving a particle from $x=0$ to $x=1$ m if the force acting on it is $F=3{x}^{2}$ N?

For the following exercises (4-5), find the mass of the one-dimensional object.

4. A car antenna that is $3$ ft long (starting at $x=0)$ and has a density function of $\rho (x)=3x+2$ lb/ft
5. A pencil that is $4$ in. long (starting at $x=2)$ and has a density function of $\rho (x)=5\text{/}x$ oz/in.

For exercises 6-8, find the mass of the two-dimensional object that is centered at the origin.

6. An oversized hockey puck of radius $2$ in. with density function $\rho (x)={x}^{3}-2x+5$
7. A plate of radius $10$ in. with density function $\rho (x)=1+ \cos (\pi x)$
8. A disk of radius $5$ cm with density function $\rho (x)=\sqrt{3x}$

For the following exercises (9-18), solve each problem.

9. 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?
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?
11. 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?
12. For the cable in the preceding exercise, how much work is done to lift the cable $50$ ft?
13. 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?
14. The force of gravity on a mass $m$ is $F=\text{−}((GMm)\text{/}{x}^{2})$ newtons. For a rocket of mass $m=1000\text{kg},$ compute the work to lift the rocket from $x=6400$ to $x=6500$ km. (Note: $G=6×{10}^{-17}{\text{N m}}^{2}\text{/}{\text{kg}}^{2}$ and $M=6×{10}^{24}\text{kg}\text{.})$
15. A rectangular dam is $40$ ft high and $60$ ft wide. Compute the total force $F$ on the dam when
    1. the surface of the water is at the top of the dam and
    2. the surface of the water is halfway down the dam.
16. Find the work required to pump all the water out of the cylinder in the preceding exercise if the cylinder is only half full.
17. 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.
18. A cone-shaped tank has a cross-sectional area that increases with its depth: $A=(\pi {r}^{2}{h}^{2})\text{/}{H}^{3}.$ Show that the work to empty it is half the work for a cylinder with the same height and base.

Moments and Centers of Mass

For the following exercises (1-3), calculate the center of mass for the collection of masses given.

1. ${m}_{1}=1$ at ${x}_{1}=-1$ and ${m}_{2}=3$ at ${x}_{2}=2$
2. Unit masses at $(x,y)=(1,0),(0,1),(1,1)$
3. ${m}_{1}=1$ at $(1,0)$ and ${m}_{2}=3$ at $(2,2)$

For the following exercises (4-8), compute the center of mass $\overline{x}.$

4. $\rho ={x}^{2}$ for $x\in (0,L)$
5. $\rho = \sin x$ for $x\in (0,\pi )$
6. $\rho ={e}^{x}$ for $x\in (0,2)$
7. $\rho =x \sin x$ for $x\in (0,\pi )$
8. $\rho =\text{ln}x$ for $x\in (1,e)$

For the following exercise, compute the center of mass $(\overline{x},\overline{y}).$ Use symmetry to help locate the center of mass whenever possible.

9. $\rho =3$ in the triangle with vertices $(0,0),$ $(a,0),$ and $(0,b)$

For the following exercises (10-13), use a calculator to draw the region, then compute the center of mass $(\overline{x},\overline{y}).$ Use symmetry to help locate the center of mass whenever possible.

10. The region bounded by $y= \cos (2x),$ $x=-\frac{\pi }{4},$ and $x=\frac{\pi }{4}$
11. The region between $y=\frac{5}{4}{x}^{2}$ and $y=5$
12. The region bounded by $y=0,$ $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1$
13. The region bounded by $y={x}^{2}$ and $y={x}^{4}$ in the first quadrant

For the following exercises (14-15), use the theorem of Pappus to determine the volume of the shape.

14. Rotating $y=mx$ around the $y$-axis between $x=0$ and $x=1$
15. A general cylinder created by rotating a rectangle with vertices $(0,0),$ $(a,0),(0,b),$ and $(a,b)$ around the $y$-axis. Does your answer agree with the volume of a cylinder?

For the following exercises (16-18), use a calculator to draw the region enclosed by the curve. Find the area $M$ and the centroid $(\overline{x},\overline{y})$ for the given shapes. Use symmetry to help locate the center of mass whenever possible.

16. Quarter-circle: $y=\sqrt{1-{x}^{2}},$ $y=0,$ and $x=0$
17. Lens: $y={x}^{2}$ and $y=x$
18. Half-ring: ${y}^{2}+{x}^{2}=1,$ ${y}^{2}+{x}^{2}=4,$ and $y=0$

For the following exercises (19-20), solve each problem.

19. Find the generalized center of mass between $y={a}^{2}-{x}^{2},$ $x=0,$ and $y=0.$ Then, use the Pappus theorem to find the volume of the solid generated when revolving around the $y$-axis.
20. Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius $a$ is positioned with the left end of the circle at $x=b,$ $b>0,$ and is rotated around the $y$-axis.