Chapter 28

A certain spring has a free length of 12.0 in. and a spring constant of \(50.016 /\) in. How much work is required to stretch the spring from a length of 14.0 in. to 16.0 in.?

Find the centroid of four particles of equal mass located at (0,0),(4,2),(3,-5) and (-2,-3)

A spring whose free length is 10.0 in. has a spring constant of \(12.0 \mathrm{lb} / \mathrm{in} .\) Find the work needed to stretch this spring from 12.0 in. to 15.0 in.

Find the length of each curve. Evaluate or check each integral by calculator or by rule. Round approximate answers to three significant digits. \(y=(1 / 6) x^{2} \quad\) from the origin to the point \((4,8 / 3)\)

A spring has a spring constant of \(8.0 \mathrm{lb} / \mathrm{in} .\) and a free length of 5.0 in. Find the work required to stretch it from 6.0 in. to 8.0 in.

Find the specified coordinate(s) of the centroid of each area. bounded by \(y=2 \sqrt{x}\) and \(x=4 ;\) find \(\bar{x}\) and \(\bar{y}\)

Find the length of each curve. Evaluate or check each integral by calculator or by rule. Round approximate answers to three significant digits. \(y=(1 / 4) x^{2}\) from \(x=0\) to 4

Find the specified coordinate(s) of the centroid of each area. bounded by \(y=\sqrt{2 x}\) and \(x=5 ;\) find \(\bar{x}\) and \(\bar{y}\)

Find the length of each curve. Evaluate or check each integral by calculator or by rule. Round approximate answers to three significant digits. \(y=\frac{x^{3 / 2}}{\sqrt{2}}\) from \(x=0\) to 10

Find the length of each curve. Evaluate or check each integral by calculator or by rule. Round approximate answers to three significant digits. The arch of the parabola \(y=4 x-x^{2}\) that lies above the \(x\) axis

Find the length of each curve. Evaluate or check each integral by calculator or by rule. Round approximate answers to three significant digits. \(x=2 y^{2} \quad\) from \(y=0\) to 2

Find the polar moment of inertia of the volume formed when a first-quadrant area with the following boundaries is rotated about the \(x\) axis. bounded by \(y=x, x=2,\) and the \(x\) axis

Find the length of each curve. Evaluate or check each integral by calculator or by rule. Round approximate answers to three significant digits. \(x=(1 / 8) y^{2}\) from \(y=0\) to 4

$$y=3 x^{2} \quad \text { from } x=0 \text { to } 5$$

Find the polar moment of inertia of the volume formed when a first-quadrant area with the following boundaries is rotated about the \(x\) axis. bounded by \(y=x+1,\) from \(x=1\) to \(2,\) and the \(x\) axis

Find the length of each curve. Evaluate or check each integral by calculator or by rule. Round approximate answers to three significant digits. \(x=2 y^{2 / 3} \quad\) from \(y=0\) to 8

Find the polar moment of inertia of the volume formed when a first-quadrant area with the following boundaries is rotated about the \(x\) axis. bounded by the curve \(y=x^{2},\) the line \(x=2,\) and the \(x\) axis

$$y=4-x^{2} \quad \text { from } x=0 \text { to } 2$$

Find the polar moment of inertia of the volume formed when a first-quadrant area with the following boundaries is rotated about the \(x\) axis. bounded by \(\sqrt{x}+\sqrt{y}=2\) and the coordinate axes

$$y=24-x^{2} \quad \text { from } x=2 \text { to } 4$$

Find the surface area of a sphere by rotating the curve \(x^{2}+y^{2}=r^{2}\) about a diameter.

Find the polar moment of inertia of each solid with respect to its axis in terms of the total mass \(M\) of the solid. a sphere of radius \(r.\)

Find the specified coordinate(s) of the centroid of each area. bounded by \(\left.y=x^{2}-2 x-3 \text { and } y=6 x-x^{2}-3 \text { (area }=21.33\right) ;\) find \(\bar{x}\) and \(\bar{y}.\)

Writing: Find and tabulate the moments of inertia of various shapes from a structural engineering handbook or by surfing the Web. Include simple shapes as well as structural members like Ells, I-beams, and so forth. Which shapes have the greatest moment of inertia in the vertical direction? Summarize your findings in a short report.

Find the distance from the origin to the centroid of each volume. formed by rotating the first-quadrant area under the curve \(y^{2}=4 x,\) from \(x=0\) to \(1,\) about the \(x\) axis.

Find the distance from the origin to the centroid of each volume. formed by rotating the area bounded by \(y^{2}=4 x, y=6,\) and the \(y\) axis about the \(y\) axis.