Physical Applications
For the following exercises (1-3), find the work done.
- How much work is done when a person lifts a 5050 lb box of comics onto a truck that is 33 ft off the ground?
- Find the work done when you push a box along the floor 22 m, when you apply a constant force of F=100N.F=100N.
- What is the work done moving a particle from x=0x=0 to x=1x=1 m if the force acting on it is F=3x2F=3x2 N?
For the following exercises (4-5), find the mass of the one-dimensional object.
- A car antenna that is 33 ft long (starting at x=0)x=0) and has a density function of ρ(x)=3x+2ρ(x)=3x+2 lb/ft
- A pencil that is 44 in. long (starting at x=2)x=2) and has a density function of ρ(x)=5/xρ(x)=5/x oz/in.
For exercises 6-8, find the mass of the two-dimensional object that is centered at the origin.
- An oversized hockey puck of radius 22 in. with density function ρ(x)=x3−2x+5ρ(x)=x3−2x+5
- A plate of radius 1010 in. with density function ρ(x)=1+cos(πx)ρ(x)=1+cos(πx)
- A disk of radius 55 cm with density function ρ(x)=√3xρ(x)=√3x
For the following exercises (9-18), solve each problem.
- A spring has a natural length of 1010 cm. It takes 22 J to stretch the spring to 1515 cm. How much work would it take to stretch the spring from 1515 cm to 2020 cm?
- A spring requires 55 J to stretch the spring from 88 cm to 1212 cm, and an additional 44 J to stretch the spring from 1212 cm to 1414 cm. What is the natural length of the spring?
- A force of F=20x−x3F=20x−x3 N stretches a nonlinear spring by xx meters. What work is required to stretch the spring from x=0x=0 to x=2x=2 m?
- For the cable in the preceding exercise, how much work is done to lift the cable 5050 ft?
- A pyramid of height 500500 ft has a square base 800800 ft by 800800 ft. Find the area AA at height h.h. If the rock used to build the pyramid weighs approximately w=100lb/ft3,w=100lb/ft3, how much work did it take to lift all the rock?
- The force of gravity on a mass mm is F=−((GMm)/x2)F=−((GMm)/x2) newtons. For a rocket of mass m=1000kg,m=1000kg, compute the work to lift the rocket from x=6400x=6400 to x=6500x=6500 km. (Note: G=6×10−17N m2/kg2G=6×10−17N m2/kg2 and M=6×1024kg.)M=6×1024kg.)
- A rectangular dam is 4040 ft high and 6060 ft wide. Compute the total force FF on the dam when
- the surface of the water is at the top of the dam and
- the surface of the water is halfway down the dam.
- Find the work required to pump all the water out of the cylinder in the preceding exercise if the cylinder is only half full.
- A cylinder of depth HH and cross-sectional area AA stands full of water at density ρ.ρ. Compute the work to pump all the water to the top.
- A cone-shaped tank has a cross-sectional area that increases with its depth: A=(πr2h2)/H3.A=(πr2h2)/H3. 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.
- m1=1m1=1 at x1=−1x1=−1 and m2=3m2=3 at x2=2x2=2
- Unit masses at (x,y)=(1,0),(0,1),(1,1)(x,y)=(1,0),(0,1),(1,1)
- m1=1m1=1 at (1,0)(1,0) and m2=3m2=3 at (2,2)(2,2)
For the following exercises (4-8), compute the center of mass ¯x.¯¯¯x.
- ρ=x2 for x∈(0,L)
- ρ=sinx for x∈(0,π)
- ρ=ex for x∈(0,2)
- ρ=xsinx for x∈(0,π)
- ρ=lnx for x∈(1,e)
For the following exercise, compute the center of mass (¯x,¯y). Use symmetry to help locate the center of mass whenever possible.
- ρ=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 (¯x,¯y). Use symmetry to help locate the center of mass whenever possible.
- The region bounded by y=cos(2x), x=−π4, and x=π4
- The region between y=54x2 and y=5
- The region bounded by y=0, x24+y29=1
- The region bounded by y=x2 and y=x4 in the first quadrant
For the following exercises (14-15), use the theorem of Pappus to determine the volume of the shape.
- Rotating y=mx around the y-axis between x=0 and x=1
- 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 (¯x,¯y) for the given shapes. Use symmetry to help locate the center of mass whenever possible.
- Quarter-circle: y=√1−x2, y=0, and x=0
- Lens: y=x2 and y=x
- Half-ring: y2+x2=1, y2+x2=4, and y=0
For the following exercises (19-20), solve each problem.
- Find the generalized center of mass between y=a2−x2, x=0, and y=0. Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.
- 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.