Chapter 6: Applications of Double Integration
Section 6.6: Second Moments
Second moments for a lamina that occupies a plane region R, and that has density ρ, are called moments of inertia, and are defined by the integrals in Table 6.6.1.
Ix= ∫∫Rρ y2 dA
Iy= ∫∫Rρ x2 dA
Table 6.6.1 Moments of inertia
These quantities are called "second" moments because the distance between an element of mass and a coordinate axis is squared. For Ix, the second moment about the x-axis, the element of mass is at a distance y from that axis; for Iy, the second moment about the y-axis, it is at a distance x from the axis.
If the total mass m=∫∫Rρ dA were concentrated at one point, equivalent moments of inertia would be generated if that point were at distances Rx=Iy/m from the x-axis and Ry=Ix/m from the y-axis. These distances are called the radii of gyration.
Torque, Rotational Inertia, and Second Moments
What is the physical meaning of a mass times a distance squared? Why is such a product of interest in the applications?
For straight-line motion, the mass m of a particle is the constant of proportionality between the applied force F, and the resulting acceleration a, a relationship captured in Newton's second law F=m a.
This section will show that m r2 is the constant of proportionality between an applied torque, and a resulting angular acceleration. (A force F applied at a distance r from a pivot point generates a torque (or twist) defined as τ=R×F, where R is a vector from the pivot point to the point where the force is applied.)
Let an element of mass dm rotate about the x-axis at a distance r from the axis. Measure the angle through which the element has turned by θt. Then the quantities in Table 6.6.2 are relevant.
Arc length: st
Rim (or linear) speed: s.t
Linear acceleration on rim: a=s..t
Table 6.6.2 Rotation of dm about x-axis
If r=R and F=F, then for a force F tangential to the circle through which dm rotates,
τ = R×F = R F sinπ/2 = R F = r F
Multiply the scalar form of Newton's second law by r to obtain
=r m a
=r m s..t
=r m r θ..t
=m r2 θ..t
The constant of proportionality between τ, the magnitude of the torque, and the resulting angular acceleration θ..t is the scalar m r2. This quantity measures the "rotational inertia" and is the content of the integrals that define Ix and Iy, the moments of inertia about the x- and y-axes, respectively.
For the lamina described in each example below, find the moments of inertial Ix and Iy, the total mass m, and the radii of gyration Rx and Ry. In Examples 6.6.1, 6.6.3, and 6.6.9 where no density is given, set ρ=1. The lamina in each example below is taken from the corresponding example in Section 6.6.5.
The lamina occupies R, the region bounded by y=1−x2 and the x-axis.
See Example 6.5.1.
The lamina has the shape of R in Example 6.6.1, and its density is equal to the square of the distance from the point 0,1/2. See Example 6.5.2.
The lamina has the shape of the triangle whose vertices are 0,0,a,0,b,c, where a,b,c are positive, and a>b. See Example 6.5.3.
The lamina has the shape of the triangle whose vertices are 0 ,0,0,3,4,0, and whose density is ρ=1+2 x+ y/2. See Example 6.5.4.
The lamina is a semicircle whose density is equal to the distance from the center of the circle, and whose radius is 1. See Example 6.5.5.
The lamina has the shape of the cardioid r=1+cosθ, with density ρ=1+2 x2+3 y2/10. See Example 6.5.6.
The lamina has density ρ=1+2 x2+3 y2/10, and is in the shape of R, the region bounded by x=y2 and y=x−2. See Example 6.5.7.
The lamina has the shape of an isosceles right triangle with legs of length 1, and has density that is twice the square of the distance from the vertex of the right angle.
Hint: Put the right angle at the origin. See Example 6.5.8.
The lamina has the shape of R, the region bounded by the graphs of x=0, and x=y−y3, where y∈0,1. See Example 6.5.9.
The lamina with density ρ=3+2 x+3 y has the shape of R, the region bounded by the ellipse x2+4 y2=1. See Example 6.5.10.
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