(a)
To write:
An integral expression for moment of inertia
Solution:
Explanation:
1) Concept:
The moment of inertia of a particle of mass m about an axis is defined to be
2) Calculations:
The given body about which the moment of inertia needs to be found is a thin sheet in the shape of a surface
And density function is
Therefore, the mass of each element is
And we have to find the moment of inertia about
The distance of each element from the
Therefore,
Moment of inertia of each element about
The moment of inertia of total sheet about
Conclusion:
An integral expression for moment of inertia
(b)
To find:
The moment of inertia about
Solution:
Explanation:
1) Concept:
Use the formula:
Where
2) Given:
Density
3) Calculations:
The given equation of cone is
Therefore,
The vector equation of surface
Differentiating with respect to
Then,
Therefore,
From concept,
The moment of inertia about
But
Therefore, above integral becomes,
Since,
Using polar coordinates,
The above integral becomes,
By integrating with respect to
By applying the limits of integration,
By simplifying,
Integrating with respect to
By applying the limits of integration,
Thus,
Conclusion:
The moment of inertia about
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- Elementary Geometry For College Students, 7eGeometryISBN:9781337614085Author:Alexander, Daniel C.; Koeberlein, Geralyn M.Publisher:Cengage,Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,