moments and products of inertia, inertia tensor
DESCRIPTION
Lecture Notes in Classical Mechanics for undergraduate students in physics. Rotation of Rigid body, Moments and products of Inertia, Inertia Tensor.TRANSCRIPT
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This lecture Note series on Classical Mechanics is for 1st year undergraduate students in Physics. Compiled by – Dr. Abhijit Kar Gupta, [email protected] Page 1
Moment of Inertia of a Rigid Body
By - Abhijit Kar Gupta, [email protected]
Consider the rotation of a rigid body.
What is a ‘rigid body’?
For any rigid body, the relative positions of the particles of it remain fixed: ( )
= const.,
where , ’s are the position vectors of the th and th particles with respect to some fixed
coordinate system. Here we have considered a system of particles having 1, 2, 3,….N particles of
masses , , ,…. respectively.
What is a ‘rotation’?
Euler’s Theorem:
In any kind of motion, where one point in the rigid body is fixed, it must be rotation.
Motion = Translation + Rotation
For a system of particles, total angular momentum is the sum of angular momenta of all
the particles:
= ∑ = ∑ = ∑ [The body is rotating with angular velocity, ]
= ∑ ( ) ∑ ( ) [Using the relation, ( ) ( ) ( )]
= ∑ ∑ ( )
Now we can find out the x-component to be the following:
∑
∑ ∑
Similarly, the y- and z-components can be separated out.
For the coefficients of , and we write,
∑
∑
, ∑ and ∑
Similarly, we can write the components of in terms of , and and that of in terms of ,
and so that we have the following relations:
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This lecture Note series on Classical Mechanics is for 1st year undergraduate students in Physics. Compiled by – Dr. Abhijit Kar Gupta, [email protected] Page 2
Where , , are the Moments of Inertia of the body about x, y and z-axes respectively. , , are
the Products of Inertia. For example, is the product which is defined with respect to the xz and yz planes,
that means in the expression of the distances ( and ) of a point particle inside the body are measured
w.r.t the xz-plane and yz-plane and then the product is calculated.
Now, we have got the following expressions for the Moments and Products of Inertia:
∑
∑
∑
∑
∑
∑
So, we see that the products of Inertia share symmetric relations among them.
Now we can think of a Matrix containing the moments and products of inertia:
(
)
Now is called the Inertia Tensor, the matrix is a symmetric matrix with six independent components.
Thus the angular momentum components can be written in a short form: ∑ , where and
.
Calculation of Kinetic Energy:
∑
=
∑ =
∑
=
∑ [ as ( ) ( ) ]
=
∑ =
∑ =
∑
=
∑ =
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This lecture Note series on Classical Mechanics is for 1st year undergraduate students in Physics. Compiled by – Dr. Abhijit Kar Gupta, [email protected] Page 3
In terms of components we can write,
=
∑ =
∑
+ +
+ + ] ………(1)
If the body is rotating around x-axis, , and
So,
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Now if only the diagonal terms in the Moment of Inertia Tensor exist, we can write
(
) so that the Kinetic Energy now becomes
+ +
].
The axes for which the products of Inertia (off diagonal terms of the Inertia Tensor) are
zero, are called principal axes of inertia.
Note:
An important property of principal axis is that if the rigid body rotates around it the
direction of the angular momentum is the same as that of the angular velocity: = .
Ellipsoid of Inertia:
From the expression (1) we can write,
+ +
+ + ] = 1
Or, (
√ )
+ (
√ )
+ (
√ )
(
√ ) (
√ ) + (
√ ) (
√ )+
(
√ ) (
√ ) = 1
+
+ + + = 1, where
√ .
The above expression is called the ellipsoid of inertia.
In terms of direction cosines, the expression for Kinetic Energy is written as
2cos + 2cos + 2cos coscos + coscos +
coscos ] 2 ,
where )cosˆcosˆcosˆ( kji .
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This lecture Note series on Classical Mechanics is for 1st year undergraduate students in Physics. Compiled by – Dr. Abhijit Kar Gupta, [email protected] Page 4
On the relations between the Principal Moments of Inertia
The Moments of Inertia of a rigid body around the three principal axes are the following:
∑ (
), ∑ (
), ∑ (
), where zkyjxiriˆˆˆ is the position
vector of a mass point of the body.
So, ∑ (
) (
) (
)
= ∑ (
) = 2∑ . [ 2222
iiii zyxr ]
2∑ …………..(1)
The above formula is true in general, whether the body is 2D or 3D, does not matter.
But in 2D, say in XY-plane, we have all iz ’s are zero.
So, ∑ (
), ∑ (
), ∑ .
Also, 222
iii yxr .
Thus, ∑ (
) (
) = 2∑
= 2
, Perpendicular axes theorem as special case of theorem (1) in 2D.