Question 16.69

Find the coordinates of the center of mass of an isosceles triangle of uniform density bounded by the \(x\) axis, \(y = ax\), and \(y = -ax+2a\).

Question 16.70

Assuming uniform density, find the coordinates of the center of mass of the semicircle \(y = \sqrt{r^2 - x^2}\), with \(y \geq 0\).

Question 16.71

Find the average of \(f(x,y)=y\sin xy \hbox{ over }D=\) \([0,\pi]\times [0,\pi]\).

Question 16.72

Find the average of \(f(x,y)=e^{x+y}\) over the triangle with vertices \((0,0), (0, 1)\), and \((1, 0)\).

Question 16.73

Find the center of mass of the region between \(y=x^2\) and \(y=x\) if the density is \(x+y\).

Question 16.74

Find the center of mass of the region between \(y=0\) and \(y=x^2\), where \(0\leq x\leq \frac{1}{2}\).

Question 16.75

A sculptured gold plate \(D\) is defined by \(0\leq x\leq 2\pi\) and \(0\leq y\leq \pi\) (centimeters) and has mass density \(\delta(x,y)= \smash{y^2\sin^2} 4x+2\) (grams per square centimeter). If gold sells for $7 per gram, how much is the gold in the plate worth?

Question 16.76

In the previous exercise, what is the average mass density in grams per square centimeter?

Question 16.77

• (a) Find the mass of the box \([0,\frac{1}{2}]\times [0,1]\times [0,2]\), assuming the density to be uniform.
• (b) Same as part (a), but with a mass density \(\delta(x,y,z)=x^2+3y^2+z+1\).

Question 16.78

Find the mass of the solid bounded by the cylinder \(x^2+y^2=2x\) and the cone \(z^2=x^2+y^2\) if the density is \(\delta =\sqrt{x^2+y^2}\).

Question 16.79

Find the mass of the solid ball of radius \(5\) with density given by \[ \delta(x,y,z) = 2x^2 + 2y^2 + 2z^2 + 1 \] assuming the center of the ball is at the origin.

338

Question 16.80

A solid disk of radius \(9\) and height \(2\) is placed at the origin, so that it can be expressed by \(x^2 + y^2 = 81\) and \(0 \leq z \leq 2\). If the disk has a density given by \[ \delta(x,y,z) = 2x^2 + 2y^2 + 2z^2 + 1, \] find its mass.

Question 16.81

Find the center of mass of the region bounded by \(x+y+z=2,x=0,y=0\), and \(z=0\), assuming the density to be uniform.

Question 16.82

Find the center of mass of the cylinder \(x^2+y^2\leq 1,1\leq z\leq 2\) if the density is \(\delta=(x^2+y^2)z^2\).

Question 16.83

Find the average value of \(\sin^2\pi\! z\cos^2 \pi\! x\) over the cube \([0,2]\times [0,4]\times [0,6]\).

Question 16.84

Find the average value of \(e^{-z}\) over the ball \(x^2+y^2+z^2\leq 1\).

Question 16.85

A solid with constant density is bounded above by the plane \(z=a\) and below by the cone described in spherical coordinates by \(\phi =k\), where \(k\) is a constant \(0<k<\pi/2\). Set up an integral for its moment of inertia about the \(z\) axis.

Question 16.86

Find the moment of inertia around the \(y\) axis for the ball \(x^2+y^2+z^2\leq R^2\) if the mass density is a constant \(\delta\).

Question 16.87

Find the gravitational potential on a mass \(m\) of a spherical planet with mass \(M=3\times 10^{26}\) kg, at a distance of \(2\times 10^8\) m from its center.

Question 16.88

Find the gravitational force exerted on a 70-kg object at the position in Exercise 19.

Question 16.89

A body \(W\) in \(xyz\) coordinates is called symmetric with respect to a given plane if for every particle on one side of the plane there is a particle of equal mass located at its mirror image through the plane.

• (a) Discuss the planes of symmetry for an automobile shell.
• (b) Let the plane of symmetry be the \(xy\) plane, and denote by \(W^+\) and \(W^-\) the portions of \(W\) above and below the plane, respectively. By our assumption, the mass density \(\delta(x,y,z)\) satisfies \(\delta(x,y,-z)=\) \(\delta(x,y,z)\). Justify the following steps: \begin{eqnarray*} \skew2\overline z \ {\cdot} \ \intop\!\!\!\intop\!\!\!\intop\nolimits_{W} \delta(x,y,z)\,{\it dx}\,{\it dy}\,{\it dz}&=&\intop\!\!\!\intop\!\!\!\intop\nolimits_{W} z\delta(x,y,z)\, {\it dx}\,{\it dy}\,{\it dz}\\[3pt] &=&\intop\!\!\!\intop\!\!\!\intop\nolimits_{W^+} z\delta(x,y,z)\,{\it dx}\,{\it dy}\,{\it dz}+ \intop\!\!\!\intop\!\!\!\intop\nolimits_{W^-} z\delta(x,y,z)\, {\it dx}\,{\it dy}\,{\it dz}\\[3pt] &=&\intop\!\!\!\intop\!\!\!\intop\nolimits_{W^+} z\delta(x,y,z)\,{\it dx}\,{\it dy}\,{\it dz}+\intop\!\!\!\intop\!\!\!\intop\nolimits_{W^+}\!\!-w\delta(u,v,-w) {\,d} u{\,d} v{\,d} w\\[2pt] &=& 0. \end{eqnarray*}
• (c) Explain why part (b) proves that if a body is symmetrical with respect to a plane, then its center of mass lies in that plane.
• (d) Derive this law of mechanics: If a body is symmetric with respect to two planes, then its center of mass lies on their line of intersection.

Question 16.90

A uniform rectangular steel plate of sides \(a\) and \(b\) rotates about its center of mass with constant angular velocity \(\omega\).

• (a) The kinetic energy equals \(\frac{1}{2}\)(mass)(velocity)\(^2\). Argue that the kinetic energy of any element of mass \(\delta\, {\it dx}\, {\it dy} (\delta=\) constant) is given by \(\delta (\omega^2/2)(x^2+y^2)\,{\it dx}\,{\it dy}\), provided the origin (0, 0) is placed at the center of mass of the plate.
• (b) Justify the formula for kinetic energy: \[ {\rm K.E.}=\intop\!\!\!\intop\nolimits_{\rm plate} \delta \frac{\omega^2}{2}(x^2+y^2)\,{\it dx}\,{\it dy}. \]
• (c) Evaluate the integral, assuming that the plate is described by the inequalities \(-a/2\leq x\leq a/2,\) \(-b/2\leq y\leq b/2\).

Question 16.91

As is well known, the density of a typical planet is not constant throughout the planet. Assume that planet C.M.W. has a radius of \(5\times 10^8\) cm and a mass density (in grams per cubic centimeter) \[ \rho(x,y,z)=\left\{ \begin{array}{l@{\quad}c} \displaystyle \frac{3\times 10^4}{r},& r\geq 10^4 \hbox{ cm},\\[6pt] 3,& r\leq 10^4 \hbox{ cm}, \end{array}\right. \] where \(r=\sqrt{x^2+y^2+z^2}\). Find a formula for the gravitational potential outside C.M.W.

Question 16.92

Let \(D\) be a region in the part of the \(xy\) plane with \(x > 0\). Assume \(D\) has uniform density, area \(A(D)\), and center of mass \(( \overline{x}, \overline{y} )\). Let \(W\) be the solid obtained by rotating \(D\) about the \(y\) axis. Show that the volume of \(W\) is given by \[ \hbox{ vol}(W) = 2 \pi \overline{x} A(D). \]

Question 16.93

Use the previous exercise to show that if a doughnut is obtained by rotating the circle \((x-a)^2 + y^2 = r^2\) about the \(y\) axis, then the volume of the doughnut is \(2 \pi^2 a r^2\).