Problem Set: Moments and Centers of Mass

For the following exercises (1-6), calculate the center of mass for the collection of masses given.

1. [latex]{m}_{1}=2[/latex] at [latex]{x}_{1}=1[/latex] and [latex]{m}_{2}=4[/latex] at [latex]{x}_{2}=2[/latex]

2. [latex]{m}_{1}=1[/latex] at [latex]{x}_{1}=-1[/latex] and [latex]{m}_{2}=3[/latex] at [latex]{x}_{2}=2[/latex]

3. [latex]m=3[/latex] at [latex]x=0,1,2,6[/latex]

4. Unit masses at [latex](x,y)=(1,0),(0,1),(1,1)[/latex]

5. [latex]{m}_{1}=1[/latex] at [latex](1,0)[/latex] and [latex]{m}_{2}=4[/latex] at [latex](0,1)[/latex]

6. [latex]{m}_{1}=1[/latex] at [latex](1,0)[/latex] and [latex]{m}_{2}=3[/latex] at [latex](2,2)[/latex]

For the following exercises (7-16), compute the center of mass [latex]\overline{x}.[/latex]

7. [latex]\rho =1[/latex] for [latex]x\in (-1,3)[/latex]

8. [latex]\rho ={x}^{2}[/latex] for [latex]x\in (0,L)[/latex]

9. [latex]\rho =1[/latex] for [latex]x\in (0,1)[/latex] and [latex]\rho =2[/latex] for [latex]x\in (1,2)[/latex]

10. [latex]\rho = \sin x[/latex] for [latex]x\in (0,\pi )[/latex]

11. [latex]\rho = \cos x[/latex] for [latex]x\in (0,\frac{\pi }{2})[/latex]

12. [latex]\rho ={e}^{x}[/latex] for [latex]x\in (0,2)[/latex]

13. [latex]\rho ={x}^{3}+x{e}^{\text{−}x}[/latex] for [latex]x\in (0,1)[/latex]

14. [latex]\rho =x \sin x[/latex] for [latex]x\in (0,\pi )[/latex]

15. [latex]\rho =\sqrt{x}[/latex] for [latex]x\in (1,4)[/latex]

16. [latex]\rho =\text{ln}x[/latex] for [latex]x\in (1,e)[/latex]

For the following exercises (17-), compute the center of mass [latex](\overline{x},\overline{y}).[/latex] Use symmetry to help locate the center of mass whenever possible.

17. [latex]\rho =7[/latex] in the square [latex]0\le x\le 1,[/latex] [latex]0\le y\le 1[/latex]

18. [latex]\rho =3[/latex] in the triangle with vertices [latex](0,0),[/latex] [latex](a,0),[/latex] and [latex](0,b)[/latex]

19. [latex]\rho =2[/latex] for the region bounded by [latex]y= \cos (x),[/latex] [latex]y=\text{−} \cos (x),[/latex] [latex]x=-\frac{\pi }{2},[/latex] and [latex]x=\frac{\pi }{2}[/latex]

For the following exercises, use a calculator to draw the region, then compute the center of mass [latex](\overline{x},\overline{y}).[/latex] Use symmetry to help locate the center of mass whenever possible.

20. [T] The region bounded by [latex]y= \cos (2x),[/latex] [latex]x=-\frac{\pi }{4},[/latex] and [latex]x=\frac{\pi }{4}[/latex]

21. [T] The region between [latex]y=2{x}^{2},[/latex] [latex]y=0,[/latex] [latex]x=0,[/latex] and [latex]x=1[/latex]

22. [T] The region between [latex]y=\frac{5}{4}{x}^{2}[/latex] and [latex]y=5[/latex]

23. [T] Region between [latex]y=\sqrt{x},[/latex] [latex]y=\text{ln}(x),[/latex] [latex]x=1,[/latex] and [latex]x=4[/latex]

24. [T] The region bounded by [latex]y=0,[/latex] [latex]\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1[/latex]

25. [T] The region bounded by [latex]y=0,[/latex] [latex]x=0,[/latex] and [latex]\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1[/latex]

26. [T] The region bounded by [latex]y={x}^{2}[/latex] and [latex]y={x}^{4}[/latex] in the first quadrant

For the following exercises, use the theorem of Pappus to determine the volume of the shape.

27. Rotating [latex]y=mx[/latex] around the [latex]x[/latex]-axis between [latex]x=0[/latex] and [latex]x=1[/latex]

28. Rotating [latex]y=mx[/latex] around the [latex]y[/latex]-axis between [latex]x=0[/latex] and [latex]x=1[/latex]

29. A general cone created by rotating a triangle with vertices [latex](0,0),[/latex] [latex](a,0),[/latex] and [latex](0,b)[/latex] around the [latex]y[/latex]-axis. Does your answer agree with the volume of a cone?

30. A general cylinder created by rotating a rectangle with vertices [latex](0,0),[/latex] [latex](a,0),(0,b),[/latex] and [latex](a,b)[/latex] around the [latex]y[/latex]-axis. Does your answer agree with the volume of a cylinder?

31. A sphere created by rotating a semicircle with radius [latex]a[/latex] around the [latex]y[/latex]-axis. Does your answer agree with the volume of a sphere?

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

32. [T] Quarter-circle: [latex]y=\sqrt{1-{x}^{2}},[/latex] [latex]y=0,[/latex] and [latex]x=0[/latex]

33. [T] Triangle: [latex]y=x,[/latex] [latex]y=2-x,[/latex] and [latex]y=0[/latex]

34. [T] Lens: [latex]y={x}^{2}[/latex] and [latex]y=x[/latex]

35. [T] Ring: [latex]{y}^{2}+{x}^{2}=1[/latex] and [latex]{y}^{2}+{x}^{2}=4[/latex]

36. [T] Half-ring: [latex]{y}^{2}+{x}^{2}=1,[/latex] [latex]{y}^{2}+{x}^{2}=4,[/latex] and [latex]y=0[/latex]

37. Find the generalized center of mass in the sliver between [latex]y={x}^{a}[/latex] and [latex]y={x}^{b}[/latex] with [latex]a>b.[/latex] Then, use the Pappus theorem to find the volume of the solid generated when revolving around the [latex]y[/latex]-axis.

38. Find the generalized center of mass between [latex]y={a}^{2}-{x}^{2},[/latex] [latex]x=0,[/latex] and [latex]y=0.[/latex] Then, use the Pappus theorem to find the volume of the solid generated when revolving around the [latex]y[/latex]-axis.

39. Find the generalized center of mass between [latex]y=b \sin (ax),[/latex] [latex]x=0,[/latex] and [latex]x=\frac{\pi }{a}.[/latex] Then, use the Pappus theorem to find the volume of the solid generated when revolving around the [latex]y[/latex]-axis.

40. Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius [latex]a[/latex] is positioned with the left end of the circle at [latex]x=b,[/latex] [latex]b>0,[/latex] and is rotated around the [latex]y[/latex]-axis.

This figure is a torus. It has inner radius of b. Inside of the torus is a cross section that is a circle. The circle has radius a.

41. Find the center of mass [latex](\overline{x},\overline{y})[/latex] for a thin wire along the semicircle [latex]y=\sqrt{1-{x}^{2}}[/latex] with unit mass.

(Hint: Use the theorem of Pappus.)