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From a uniform disc of radius R a small disc of radius R/3 is cut . The mass of the remaining portion is M . find the moment of inrtia of such a disc about the axis passing through the geometrical centre of original disc and perpendicular to the the plane of the disc .

From  a uniform disc of  radius R a small disc of radius R/3 is cut . The mass of the remaining portion is M . find the moment of inrtia of such a disc about the axis passing through the geometrical centre of original disc and perpendicular to the the plane of the disc . 

Grade:11

1 Answers

Kapil Khare
80 Points
5 years ago
Area of the remaining disc = \pi[R2 – (R2/9)]
                                        A = \pi(8R2/9)
Mass of the remaing disc = M
Mass per unit area of the material = \sigma = 9M/(8\piR2)
Mass of the part remaining = \sigma(\piR2/9) = M/8
 
So let us assume a M/8 and -M/8 plate on the part from where we removed the small disc.
Now we have a disc of mass 9M/8 with radius R and another disc of mass -M/8 with radius R/3.
 
So Moment of Inertia of larger disc about its axis =9MR2/16
Now let us assume the distance between the centers of the two disc be d.
Now, moment of inertia of smaller disc about its axis = -MR2/144
Moment of inertia of smaller disc about center of original disc = (-MR2/144) + (-Md2/8)
 
So, total moment of inertia about the centre of original disc = (9MR2/16) – (MR2/144) – (Md2/8)
                             I = (5MR2/9) – (Md2/8)

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