Chapter Contents

• Applications of Integration
• 1. Applications of the Indefinite Integral
• 2. Area Under a Curve
• 3. Area Between 2 Curves
• 4. Volume of Solid of Revolution
• 5. Centroid of an Area
• 6. Moments of Inertia
• 7. Work by a Variable Force
• 8. Electric Charges
• 9. Average Value
• Head Injury Criterion
• a. Normal Braking
• b. Crash Tests
• c. Data
• d. Severity Index
• e. Head Injury Criterion
• f. Modelling the HIC
• g. Conclusions
• 10. Force by Liquid Pressure
• 11. Arc Length of a Curve
• 12. Arc Length of a Curve in Parametric or Polar Coordinates
• Applications of Integration Problem Solver

• Comments, Questions?

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6. Moments of Inertia

by M. Bourne

The moment of inertia is a measure of the resistance of a rotating body to a change in motion.

The moment of inertia of a particle of mass m rotating about a particular point is given by:

`"Moment of inertia" = md^2`

where d is the radius of rotation.

Inertia for a Collection of Particles

If a group of particles with masses m₁, m₂, m₃, ... , m_n is rotating around a point with distances d₁, d₂, d₃, ... d_n, (respectively) from the point, then the moment of inertia I is given by:

I = m₁d₁² + m₂d₂² + m₃d₃² +... + m_nd_n²

If we wish to place all the masses at the one point (R units from the point of rotation) then

d₁ = d₂ = d₃ = ... = d_n = R and we can write:

I = (m₁ + m₂ + m₃ ... + m_n)R²

R is called the radius of gyration.

Example 1

Find the moment of inertia and the radius of gyration w.r.t. the origin (0,0) of a system which has masses at the points given:

Mass	`6`	`5`	`9`	`2`
Point	`(-3, 0)`	`(-2, 0)`	`(1, 0)`	`(8, 0)`

Answer

Moment of Inertia for Areas

We want to find the moment of inertia, I_y of the given area, which is rotating around the y-axis.

Each "typical" rectangle indicated has width dx and height y₂ − y₁, so its area is (y₂ − y₁)dx.

If k is the mass per unit area, then each typical rectangle has mass k(y₂ − y₁)dx.

The moment of inertia for each typical rectangle is [k(y₂ − y₁)dx] x², since each rectangle is x units from the y-axis.

We can add the moments of inertia for all the typical rectangles making up the area using integration:

`I_y=kint_a^bx^2(y_2-y_1)dx`

Using a similar process that we used for the collection of particles above, the radius of gyration R_y is given by:

`R_y=sqrt((I_y)/m`

where m is the mass of the area.

Example 2

For the first quadrant area bounded by the curve

`y = 1 − x^2`,

find:

a) The moment of inertia w.r.t the y axis. (I_y)

b) The mass of the area

c) Hence, find the radius of gyration

Answer

Note: For rotation about the x-axis, the formulae become:

`I_x=kint_c^(d)y^2(x_2-x_1)dy`

and

`R_x=sqrt((I_x)/m`