ISC (Commerce) Class 11CISCE
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Theorems of Perpendicular and Parallel Axes

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  • Statement of Parallel and Perpendicular Axes Theorems and Their Applications
  • Theorem of perpendicular axes
  • Theorem of parallel axes

notes

Theorem of perpendicular axis:

Perpendicular Axis Theorem: The moment of inertia of a planar body (lamina) about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with perpendicular axis and lying in the plane of the body.

Moment of inertia about x-axis, Ix = mx2

Moment of inertia about y-axis, Iy = my2

Moment of inertia about z-axis, `"I"_z= m(sqrt(x2+y2))^2`

`I_x + I_y = mx^2 + my^2`

= `m(x^2 + y^2)`

=`m(sqrt(x2+y2))`

Ix+Iy=Iz

Hence the theorem is proved

Applicable only to planar bodies.

Theorem of parallel axis:

Parallel Axis Theorem: The moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of the distance between the two parallel axes.

This theorem is applicable to a body of any shape.

Example: Given the moment of inertia of a disc of mass M and radius R about any of its diameters to be `"MR"^2/4`, find its moment of inertia about an axis normal to the disc and passing through a point on its edge

Solution:

We can apply Perpendicular axis theorem here on x axis & y axis and get I’, moment of inertia in z axis.
`"l"_"z" = "l"_"x" + "l"_"y"`, now as because of symmetry Ix & Iy are same so Iz = I’ = 2I = `("Mr"^2)/2`

Now we can apply parallel axis theorem to find I’’.
`I'' = I' + "MR"^2 = 3/2("MR"^2)`

Moment of Inertia:

Z

Body

Axis

Figure

I

1

Thin circular ring, radius R

Perpendicular to plane, at centre

MR2

2

Thin circular ring, radius R

Diameter

`"MR"^2/2`

3

Thin rod, length L

Perpendicular to rod, at mid point

`"ML"^2/12`

4

Circular disc, radius R

Perpendicular to disc at centre

`"MR"^2/2`

5

Circular disc, radius R

Diameter

`"MR"^2/4`

6

Hollow cylinder, radius R

Axis of cylinder

MR2

7

Solid cylinder, radius R

Axis of cylinder

`"MR"^2/2`

8

Solid sphere, radius R

Diameter

2 `"MR"^2/5`

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Shaalaa.com | Principle of parallel and perpendicular axes

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Principle of parallel and perpendicular axes [00:15:57]
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