Integrals: Moment of Inertia

MOMENT OF INERTIA

1. Moment of Inertia of a Lamina

If a particle of mass m is at a distance r from a given line, then mr2 is called the moment of inertia of the particle about the given line. This line is refered to as an axis. We extend this concept to a lamina which occupies a region R in the xy-plane with density ρ(x, y) at the point (x, y). We assume ρ(x, y) is continuous. We divide the region R into small rectangles with typical rectangle of area Δx · Δy and element mass ρ(x, y)  Δx · Δy.

It is also known as polar moment of Inertia.

Note that Io = Ix + Iy

It is also the M.I of the lamina about an axis perpendicular to the xy-plane. 

Note: The tendency of the lamina to resist a change in rotational motion about an axis is measured by its moment of inertial about that axis.

For example, the M.I of a wheel of a car is what makes it difficult to start or stop the rotation of the wheel.

Radius of gyration

The radius of gyration of a lamina about an axis is the number R such that MR2 = I, where M is the mass of the lamina and I is the M.I about the axis.

WORKED EXAMPLES

Example 1 

Find the moment of inertia of a uniform rectangular lamina that occupies the region R in the xy-plane given by 0 ≤ x ≤ a, 0 ≤ y ≤ b about the x-axis, the y-axis and about the origin.

Solution 

Given the rectangular lamina is of constant density ρ.

Example 2 

Find the moment of inertia of a homogeneous circular disk in the xy-plane with centre origin and radius a about the x-axis and the y-axis.

Solution 

Put y = a sin θ

⸫ dy = acos θ dθ

When y = 0, sin θ = 0 ⇒ θ = 0

When y = a, sin θ = 1  ⇒ θ = π/2

Because the region is symmetric about both the axis,

Note: So, the M.I of a circular disk of radius a about a diameter is 1/4 Ma2, where M is the mass of the lamina.

M.I is circular disk of radius a about an axis through the centre and perpendicular to it is 1/2 Ma2.

Example 3 

Find Ix, Iy, Iz of the lamina occupying the region R in the xy-plane, where R = {(x,y)|1≤x≤3,1≤ y ≤4} and ρ(x, y) = ky2

Solution 

Given the lamina occupies a rectangular region in the xy-plane.

Example 4 

A thin plate covers the triangular region bounded by the x-axis and the lenis x = 1 and y = 2x in the first quadrant. The plate's density at the point (x, y) is ρ(x, y) = 6x + 6y + 6. Find the M.I of the plate about the x-axis, about the y-axis and about the origin.

Solution 

Example 5 

Find the moment of inertia about the x-axis of a thin plate bounded by the parabola x = y - y2 and the line x + y = 0 if the density ρ(x, y) = x + y.

Solution 

Example 6 

Find the M.I of a uniform elliptic lamina about its major axis and minor axis.

Solution 

Given a uniform elliptic lamina in the xy-plane. Equation of the boundary of the region is the ellipse

Its centre is the origin, major and minor axes are the x-axis and the y-axis.

That is, M.I about an axis perpendicular to the lamina through its centre is

2. Moment of Inertia of a Solid in Space

For a solid region, the element mass at the point, P (x, y, z) is ρΔx Δy Δz, where ρ(x,y,z) is the density at P. It may be a constant in some case.

We assume ρ(x,y,z) is continuous.

WORKED EXAMPLES

Example 1 

A solid cube in the first octant is bounded by the coordinate planes and the planes x = 1, y = 1, z = 1. The density of the cube is ρ (x,y,z) = x + y + z + 1. Find the moments of inertia about the coordinate axes. 

Solution 

Given a solid cube bounded by the planes x = 0, x = 1, y = 0, y = 1, z = 0, z = 1.

Given the density at any point (x, y, z) is ρ (x, y, z) = x + y + z + 1.

Example 2 

Find Iz for the solid cylinder x2 + y2 ≤ a2, 0 ≤ z ≤ h with constant density ρ.

Solution 

Given a solid cylinder

Base circle is x2 + y2 = a2 and height is h.

Density ρ is a constant

We shall evaluate using cylindrical polar coordinates 

Note: M.I of a solid cylinder of base radius a about the axis of the cylinder = M/2 a2

Example 3 

Find the M.I of a solid sphere of radius a about a diameter.

Solution 

We shall use the basic method of slicing to sphere into circular disks perpendicular to the diameter l and use Riemann sum idea.

Consider a typical disk at a distance x from centre O and thickness Δx.

⸫ M.I of the solid sphere about the diameter l is

Example 4 

Find the M.I of a solid right circular cone about its axis.

Solution 

Consider a solid right circular cone of height h and base circle of radius a choose the vertex of the cone as the origin and its axis as the z-axis as in Fig. 5.105.

We shall use basic Riemann sum idea.

Divide the solid cone into circular disks perpendicular to the axis.

i.e., the z-axis.

Take a typical disk at a distance z from O and thickness Δz.

⸫ M.I of the disk about an axis perpendicular to it is

The following theorems are useful in finding M.I if the M.I of a body is required about an axis other than the principal axis.

Theorem 1 Parallel axis theorem (Steiner's Theorem)

If the M.I of a body of mass M about an axis through the centre of gravity is I, then the M.I of the body I' about a parallel axis at a distance d from the first axis is given by I' = I + Md2

Theorem 2 Perpendicular axis theorem

If Ix and Iy be the M.I of a plane lamina about two perpendicular axes ox, oy in its plane, then the M.I of the lamina about an axis oz through O and perpendicular to its plane is

Iz = Ix + Iy

Note: The perpendicular axis theorem is valid only for a plane lamina whereas the parallel axis theorem is valid for any rigid body.

WORKED EXAMPLES

Example 5 

Find the M.I of a solid right circular cone having base radius a and height h about (i) its axis (ii) an axis through the vertex and perpendicular to its axis (iii) a diameter of its base.

Solution 

Refer Fig. 5.105 in example 4

(i) By example 4, M.I of the cone about the axis of the cone (z-axis) is

= 3/10 Ma2

(ii) An axis through the vertex and perpendicular to it is the y-axis (of course the x-axis also)

A diameter of the disk is parallel to the y-axis.

Since the diameter of base is parallel to the diameter of the disk, by parallel axis theorem, we get

⸫ M.I of the cone about the diameter of the base

EXERCISE

1. Find the M.I about the x-axis and the y-axis of the part of the disk x2 + y2 ≤ a2 in the first quadrant and with constant density ρ.

2. Find the M.I, Ix, Iy, Io for an isosceles right triangle with equal sides a if the density at any point is proportional to the square of the distance from the vertex opposite to hypotenuse.

 [Hint: vertex at right angle as origin as in the figure]

3. Find the M.I about the x-axis of the area enclosed by the line x = 0, y = 0,

4. Find the M.I about the x-axis of a thin plate of density 1 bounded by the circle x2 + y2 = 4. Then use this result to find Iy and Io for the plate.

5. Find the M.I about the y-axis of a thin plate bounded by the x-axis, the lines x = ± 1 and the parabola y = x2 if p(x, y) = 7y + 1.

6. Find the moment of inertia about the z-axis of a homogeneous tetrahedron bounded by the planes x = 0, y = 0, z = x + y and z = 1.

7. Find the M.I of an octant of the ellipsoid about the x-axis.

ANSWERS TO EXERCISE