Calculus II
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Moments and center of mass

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Review Videos

Videos, Math Dr. Bob:

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03 Theory

Theory 1

Moment

The moment of a region to an axis is the total (integral) of mass times distance to that axis:

Moment to :

Moment to :

Notice the swap in letters

  • integrand has factor
  • integrand has factor

Notice the total mass

If you remove or factors from the integrands, the integrals give total mass .

These formulas are obtained by slicing the region into rectangular strips that are parallel to the axis in question.

The area per strip is then:

  • — region under
  • — region between and
  • — region ‘under’
  • — region between and

center

center

The idea of moment is related to:

  • Torque balance and angular inertia
  • Center of mass

The center of mass (CoM) of a solid body is a single point with two important properties:

  1. “average position” of the body
    • The average position determines an effective center of dynamics. For example, gravity acting on every bit of mass of a rigid body acts the same as a force on the CoM alone.
  2. “balance point” of the body
    • The net torque (rotational force) about the CoM, generated by a force distributed over the body’s mass, equivalently a force on the CoM, is zero.

Centroid

When the body has uniform density, then the CoM is also called the centroid.

Center of mass from moments

Coordinates of the CoM:

Here is the total mass of the body.

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04 Illustration

Example - CoM of a parabolic plate

CoM of a parabolic plate

Find the CoM of the region depicted:

center

Solution

(1) Compute the total mass:

Area under the curve with density factor :

(2) Compute :

Formula:

Interpret and calculate:

(3) Compute :

Formula:

Width of horizontal strips between the curves:

Interpret :

Calculate integral:

(4) Compute CoM coordinates from moments:

CoM formulas:

Insert data:

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05 Theory

Theory 2

A downside of the technique above is that to find we needed to convert the region into functions in . This would be hard to do if the region was given as the area under a curve but cannot be found analytically. An alternative formula can help in this situation.

Midpoint of strips for opposite variables

When the region lies between and , we can find with an -integral:

When the region lies between and , we can find with a -integral:

Region under a curve

For the region “under the curve” , just set:

For the region “under the curve” , set:

The idea for these formulas is to treat each vertical strip as a point concentrated at the CoM of the vertical strip itself.

center

The height to this midpoint is , and the area of the strip is , so the integral becomes .

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06 Illustration

Example - Computing CoM using only vertical strips

Computing CoM using only vertical strips

Find the CoM of the region:

center

Solution

(1) Compute the total mass :

Area under the curve times density :

(2) Compute using vertical strips:

Plug into formula:

Integration by parts. Set , and so , :

(3) Compute , also using vertical strips:

Plug and into formula:

Integration by ‘power to frequency conversion’:

(4) Compute CoM:

CoM formulas:

Plug in data:

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07 Theory

Theory 3

Two useful techniques for calculating moments and (thereby) CoMs:

  • Additivity principle
  • Symmetry

Additivity says that you can add moments of parts of a region to get the total moment of the region (to a given axis).

A symmetry principle is that if a region is mirror symmetric across some line, then the CoM must lie on that line.

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08 Illustration

Example - Center of mass using moments and symmetry

Center of mass using moments and symmetry

Find the center of mass of the two-part region:

center

Solution

(1) Symmetry: CoM on -axis

Because the region is symmetric in the -axis, the CoM must lie on that axis. Therefore .

(2) Additivity of moments:

Write for the total -moment (distance measured to the -axis from above).

Write and for the -moments of the triangle and circle.

Additivity of moments equation:

(3) Find moment of the circle :

By symmetry we know .

By symmetry we know .

Area of circle with is , so total mass is .

Centroid-from-moments equation:

Solve the equation for :

(4) Find moment of the triangle using integral formula:

Similar triangles:

center Quick linear interpolation function:

Thus:

Conclude:

(5) Apply additivity:

(6) Total mass of region:

Area of circle is . Area of triangle is . Thus .

(7) Compute center of mass from total and total :

We have and . Plug into formula:

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Example - Center of mass - two part region

Center of mass - two part region

Find the center of mass of the region which combines a rectangle and triangle (as in the figure) by computing separate moments. What are those separate moments? Assume the mass density is .

center

Solution

(1) Apply symmetry to rectangle:

By symmetry, the center of mass of the rectangle is located at .

Thus and .

(2) Find moments of the rectangle:

Total mass of rectangle . Thus:

(3) Find moments of the triangle:

Area of vertical slice . Distance from -axis . Total integral:

Total integral:

(4) Add up total moments:

General formulas: and

Plug in data: and

(5) Find center of mass from moments:

Total mass of triangle .

Total combined mass .

Apply moment relation:

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Improper integrals

Videos

Review Videos

Videos, Math Dr. Bob:

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03 Theory

Theory 1

Improper integrals are those for which either a bound or the integrand itself become infinite somewhere on the interval of integration.

Examples:

(a) the upper bound is (b) the integrand goes to as (c) the integrand is at the point

The limit interpretation of (a) is this:

The limit interpretation of (b) is this:

The limit interpretation of (c) is this:

These limits are evaluated using the usual methods.

An improper integral is said to be convergent or divergent according to whether it may be assigned a finite value through the appropriate limit interpretation.

For example, converges while diverges.

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04 Illustration

Example - Improper integral - infinite bound

Improper integral - infinite bound

Show that the improper integral converges. What is its value?

Solution

(1) Improper integral definition:

(2) Replace infinity with a new symbol :

(3) Take limit as :

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Improper integral - infinite integrand

Improper integral - infinite integrand

Show that the improper integral converges. What is its value?

Solution

(1) Improper integral definition:

(2) Switch to and integrate:

(3) Take limit as :

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Example - Improper integral - infinity inside the interval

Example - Improper integral - infinity inside the interval

Does the integral converge or diverge?

Solution

(1) WRONG APPROACH:

It is tempting to compute the integral incorrectly, like this:

But this is wrong! There is an infinite integrand at . We must break it into parts!

(2) Break apart at discontinuity:

(3) Improper integral definition, both terms separately:

(3) Integrate:

(4) Limits:

Neither limit is finite. For to exist we’d need both of these limits to be finite. So the original integral diverges.

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05 Theory

Theory 2

Two tools allow us to determine convergence of a large variety of integrals. They are the comparison test and the -integral cases.

Comparison test - integrals

The comparison test says:

  • When an improper integral converges, every smaller integral converges.
  • When an improper integral diverges, every bigger integral diverges.

Here, smaller and bigger are comparisons of the integrand at all values (accounting properly for signs), and the bounds are assumed to be the same.

For example, converges, and implies (when ), therefore the comparison test implies that converges.

-integral cases

Assume and . We have:

Proving the -integral cases

It is easy to prove the convergence / divergence of each -integral case using the limit interpretation and the power rule for integrals. (Or for , using .)

Additional improper integral types

The improper integral also has a limit interpretation:

The double improper integral has this limit interpretation:

Where is any finite number. This double integral does not exist if either limit does not exist for any value of .

Double improper is not simultaneous!

Watch out! This may happen:

This simultaneous limit might exist only because of internal cancellation in a case where the separate individual limits do not exist! We do not say ‘convergent’ in these cases!

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06 Illustration

Example - Comparison to -integrals

Comparison to p-integrals

Determine whether the integral converges:

(a)

(b)

Solution

(a) (1) Observe large tendency:

Consider large values. Notice the integrand tends toward for large .

(2) Try comparison to :

Validate. Notice and when .

(3) Apply comparison test:

We know:

We conclude:

(b) (1) Observe large tendency:

Consider large values. Notice the integrand tends toward for large .

(2) Try comparison to :

Validate. Notice and when .

(3) Apply comparison test:

We know:

We conclude:

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