Chapter 15 1 Find Its Center Of mass The Density Any

Stewart_Calc_7ET ch15sec05

MULTIPLE CHOICE

1. Find the center of mass of a lamina in the shape of an isosceles right triangle with equal

sides of length if the density at any point is proportional to the square of the distance

from the vertex opposite the hypotenuse. Assume the vertex opposite the hypotenuse is

located at , and that the sides are along the positive axes.

a.

b.

c.

d.

e.

None of these

2. Find the mass and the center of mass of the lamina occupying the region R, where R is the

triangular region with vertices and , and having the mass density

a.

,

b.

,

c.

,

d.

,

3. An electric charge is spread over a rectangular region

Find the total charge on R if the charge density at a point in R (measured in coulombs

per square meter) is

a.

coulombs

b.

coulombs

c.

coulombs

d.

coulombs

25

804

91

300

265

4. Find the center of mass of the lamina that occupies the region D and has the given density

function, if D is bounded by the parabola and the x-axis.

a.

b.

c.

d.

e.

None of these

5. Find the mass of the lamina that occupies the region D and has the given density function, if

D is bounded by the parabola and the line .

a.

b.

c.

27

d.

e.

None of these

NUMERIC RESPONSE

1. Find the mass of the lamina that occupies the region and has the given density function.

Round your answer to two decimal places.

3

2

27

2

2. A lamina occupies the part of the disk in the first quadrant. Find its center of

mass if the density at any point is proportional to its distance from the x-axis.

SHORT ANSWER

1. Find the center of mass of the system comprising masses mk located at the points Pk in a

coordinate plane. Assume that mass is measured in grams and distance is measured in

centimeters.

m1 = 4, m2 = 3, m3 = 2

P1(–3, –3), P2(0, 3), P3(–2, –1)

2. Find the center of mass of the lamina of the region shown if the density of the circular

lamina is four times that of the rectangular lamina.

2–2 x

1

–1

y

3. Find the mass and the center of mass of the lamina occupying the region R, where R is the

region bounded by the graphs of the equations and and having the

mass density

4. Find the mass and the center of mass of the lamina occupying the region R, where R is the

region bounded by the graphs of and and having the mass

density

5. Find the mass and the moments of inertia and and the radii of gyration and

for the lamina occupying the region R, where R is the rectangular region with vertices

and , and having uniform density

6. Find the mass and the moments of inertia and and the radii of gyration and

for the lamina occupying the region R, where R is the region bounded by the graphs of the

equations and and having the mass density