Any help is appreicated! 1. Consider a thin constant-density rectangular plate i

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Any help is appreicated!

1. Consider a thin constant-density rectangular plate in the xy-plane with side lengths a and b centered at the origin and suppose it rotates about the y-axis (Figure 2). Note that a point

(x, y) in the region has a distance from the

y-axis of r = |x|, which means r2 = x2. Explain why the moment of inertia is

Repeat the calculations of Steps 1 and 2 when the axis of rotation of the rectangular plate is along an edge of length b. Show that I = ma2/3.

Comparing the results of Steps 2 and 3, does the plate have more resistance to rotation when the axis of rotation is through the center of the plate or along an edge?

b/2 a/2
I= x2dxdy

Figure2
2. Evaluate the integral in Step 1 to show that I = a3b/12. Now note that the mass of the plate is

b/2 a/2
m = density × area = ab. Show that the moment of inertia is I = ma2/12.

5. Now suppose that a thin constant-density rectangular plate in the xy-plane with side lengths a and b centered at the origin is rotated about the z-axis (Figure 3). Note that the distance squared between a point (x, y) in the region and the z-axis is x2 + y2. Show that the moment of inertia is

7. Now let’s work in polar coordinates. Consider a flat constant-density circular plate of radius R. Show that if the plate is rotated about an axis perpendicular to the plate through the center of the plate (Figure 4) then

9. Consider a constant-density flat circular plate of radius R. Show that if the plate is rotated about a diameter (Figure 5) then I = mR2/4. Explain why the moment of inertia for a flat circular plate is greater if the axis of rotation is perpendicular to the plate and passes through the center (Step 8) than if it lies on a diameter (Step 9)?

b/2 a/2 b/2 a/2

I =
6. Evaluate the integral in Step 5 to show that

(x2 + y2 ) dxdy . I = ab(a2 + b2)/12 = m(a2 + b2)/12.

Figure 3

2 R 00

I =
8. Evaluate the integral in Step 7 and note that

r3 drd . m = R2 to show that I = mR2/2.

Figure 4

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Figure 5

10. With these two-dimensional calculations complete, let’s move up to three dimensions – first in Cartesian coordinates. Consider a constant-density box centered at the origin with sides of length a parallel to the x-axis, sides of length b parallel to the y-axis, and sides of length c parallel to the z-axis (Figure 6). First consider the moment of inertia with respect to rotation about the z-axis. Explain why

Suppose the box in Step 10 is rotated about one of the edges parallel to the z-axis? Show that I = m(a2 + b2)/3. (The calculation is easier if one corner of the box is placed at the origin.)

Show that the moment of inertia of a solid circular cylinder rotated around an axis perpendicular to the base of the cylinder and passing through the center of the base is I = mR2/2, where R is the radius of the base. Why is I independent of the length of the cylinder?

Show that the moment of inertia of a solid cone rotated about an axis perpendicular to the base of the cone and passing through the center of the base is I = 3mR2/10, where R is the radius of the base.

Now use spherical coordinates to show that the moment of inertia of a solid sphere of radius R rotated around an axis passing through the center of the sphere is given by

I = 2 R 4 s i n 3 d d d 000

Evaluate the integral in Step 15 and use the fact that the mass is m = (4R3/3) to show that I = 2mR2/5.

Finally consider a thin spherical shell of radius R. Repeat the calculation of Step 16 except note that = R for all and . Also use the fact that the surface area of a sphere is 4R2 and that is an area density. Show that the moment of inertia of a spherical shell is I = 2mR2/3.

Suppose a figure skater begins a spin with a certain amount of rotational kinetic energy 1 I2 that remains 2

constant. Explain why she spins faster (her angular speed increases) when she pulls her arms against her body.

Explanation / Answer

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