Concept explainers
The inertia tensor for a uniform, thin hollow cone such as an ice cream cone of mass
Answer to Problem 10.27P
The moment of inertia tensor of the hollow cone which spins about its pointed end is
Explanation of Solution
Write the moment of inertia of a body about an axis.
Here,
The moment of inertia of a body about an axis depends on the mass and the distance of the mass distribution from the axis of rotation.
Consider a hollow cone which is spinning about the
Here,
According to the given figure, the polar coordinates of any point on a cone in three dimensional space is,
Here,
From the above figure,
The volume of small element of cone I cylindrical coordinate is,
Use equation (V) and (III) in equation (II),
Use equation (IV) in the above equation,
The mass of the cone is given by,
Use equation (V)in equation (VI),
Substitute
Solve above equation for
Use equation (IX) in equation (VI),
He moment of inertia of the cone about the x axis is,
Use equation (IV) and (V) in the above equation,
Using equation (V),
Use equation (IX) in the above equation,
By the symmetry of the axes, the moment of inertia
The moment of inertia about its pointed end of a hollow cone is,
Use equation (XI) (XII) and (X) in equation above,
Conclusion:
Therefore, The moment of inertia tensor of the hollow cone which spins about its pointed end is
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