(a) Prove the theorem of perpendicular axes. (Hint : Square of the distance of a point (x, y) in the x–y plane from an axis through the origin and perpendicular to the plane is x2+y2). (b) Prove the theorem of parallel axes. (Hint : If the centre of mass of a system of n particles is chosen to be the origin (∑ m _t r_t = 0)
(a)
Let the point mass d m be placed at point P in 3-D space and it’s positional coordinates are (x, y, z) as shown in the figure.
The expression for the moment of inertia about the x-axis is,
I x = ∫ d m x 2
Here, x is the coordinate of the point mass along the x-axis.
The expression for the moment of inertia about the y-axis is,
I y = ∫ d m y 2
Here, y is the coordinate of the point mass along the y-axis.
The expression for the moment of inertia about the z-axis is,
I z = ∫ d m z 2
Here, z is the coordinate of the point mass along the z-axis.
The geometric relationship between the x-coordinate, y-coordinate and z-coordinate is,
z 2 = x 2 + y 2
The sum of the moment of inertia about the x-axis and y-axis is,
I x + I y = ∫ d m x 2 + ∫ d m y 2 = ∫ d m ( x 2 + y 2 )
Substituting the value of ( x 2 + y 2 ) in the above equation, we get:
I x + I y = ∫ d m ( z 2 ) I x + I y = ∫ d m z 2
Substituting I z for ∫ d m z 2 in the above equation, we get:
I x + I y = I z
According to the theorem of perpendicular axes, the sum of the moment of inertia of any mass system about two mutually perpendicular axes is equal to the moment of inertia of the mass system about the third perpendicular axis.
Thus, the theorem of perpendicular axes is proved.
(b)
Let a point mass d m be at a distance r from the axis passing through the centre of mass of the whole-mass system as shown in the figure.
The expression for the moment of inertia of the mass system about the axis passing through the centre of mass of the system is,
I O = I C M = ∫ ( d m ) r 2
The expression for the moment of inertia of the mass system about any given axis CD which is parallel to the centroidal axis is,
I C D = ∫ d m ( d + r ) 2 = ∫ d m ( r 2 + d 2 + 2 r d ) = ∫ ( d m ) r 2 + ∫ ( d m ) d 2 + ∫ d m ⋅ 2 r d = ∫ ( d m ) r 2 + ∫ ( d m ) d 2 + 2 ∫ ( d m ) r d …… (1)
Net moment of inertia of all the particles about the centroidal axis is zero. Therefore,
∫ ( d m ) r d = 0
Substituting the above value in equation (1), we get:
I C D = ∫ ( d m ) r 2 + ∫ ( d m ) d 2 + 2 × ( 0 ) = ∫ ( d m ) r 2 + ∫ ( d m ) d 2 + 0 = ∫ ( d m ) r 2 + ∫ ( d m ) d 2
Substituting I C M for ∫ ( d m ) r 2 in the above equation, we get:
I C D = I C M + ∫ ( d m ) d 2
Let M be the mass of a rigid body, then according to the above relationship,
I C D = I C M + M d 2
According to the parallel axes theorem, the moment of inertia of any mass system about any given axis is equal to the sum of the moment of inertia about the centroidal axis parallel to the given axis and the product of mass and the square of the distance between the two parallel axes.
Thus, the theorem of parallel axes is proved.
(a)
Let the point mass
The expression for the moment of inertia about the x-axis is,
Here,
The expression for the moment of inertia about the y-axis is,
Here,
The expression for the moment of inertia about the z-axis is,
Here,
The geometric relationship between the x-coordinate, y-coordinate and z-coordinate is,
The sum of the moment of inertia about the x-axis and y-axis is,
Substituting the value of
Substituting
According to the theorem of perpendicular axes, the sum of the moment of inertia of any mass system about two mutually perpendicular axes is equal to the moment of inertia of the mass system about the third perpendicular axis.
Thus, the theorem of perpendicular axes is proved.
(b)
Let a point mass
The expression for the moment of inertia of the mass system about the axis passing through the centre of mass of the system is,
The expression for the moment of inertia of the mass system about any given axis CD which is parallel to the centroidal axis is,
Net moment of inertia of all the particles about the centroidal axis is zero. Therefore,
Substituting the above value in equation (1), we get:
Substituting
Let M be the mass of a rigid body, then according to the above relationship,
According to the parallel axes theorem, the moment of inertia of any mass system about any given axis is equal to the sum of the moment of inertia about the centroidal axis parallel to the given axis and the product of mass and the square of the distance between the two parallel axes.
Thus, the theorem of parallel axes is proved.
