Moment of Inertia - Hollow Objects

[JerS]

Homework Statement

1. Given that the moment of inertia of a thin conical shell, of base radius a and height 3a, about
an axis through its apex perpendicular to its symemtry axis is 19
4 ma2, and that the centre of
mass of the shell is along its symmetry axis a distance a from the base and 2a from the apex,
find the moment of inertial of the conical shell about an axis which passes through its centre
of mass and is perpendicular to its symmetry axis.
2. A partially hollowed out thick uniform spherical shell, of mass m and outer radius a, has
moment of inertia 23
15 ma2 about a tangent. What is the moment of inertia of the shell about a
diameter, given that its centre of mass is at the centre of the shell?

Homework Equations

I = Sigma m (r^2)

The Attempt at a Solution

Lots on paper



Thanks tons, again this is for an exam tomorrow

 

[tiny-tim]

Welcome to PF!

Hi JerS! Welcome to PF!

(try using the X² tag just above the Reply box )

Hint: use the parallel axis theorem …

what do you get? ​

 

[nlightner]

I would first restate the problem and make sure I fully understand the given information and what is being asked.

1. The problem is asking for the moment of inertia of a thin conical shell about an axis which passes through its center of mass and is perpendicular to its symmetry axis. The given information includes the moment of inertia of the shell about an axis through its apex perpendicular to its symmetry axis, the dimensions of the shell, and the location of its center of mass.

2. The second problem is asking for the moment of inertia of a partially hollowed out thick spherical shell about a diameter, given its mass, outer radius, and the location of its center of mass. The given information also includes the moment of inertia of the shell about a tangent.

To solve these problems, I would use the equation for moment of inertia, I = Σmr^2, where m is the mass of each small element of the object and r is the distance of that element from the axis of rotation.

For the first problem, I would use the given moment of inertia and the dimensions of the shell to calculate the mass of the shell (m). Then, I would use the given location of the center of mass to find the distance of each small element from the axis of rotation (r). Finally, I would plug these values into the equation for moment of inertia to find the moment of inertia about the desired axis.

For the second problem, I would use the given moment of inertia about a tangent and the dimensions of the shell to calculate the mass of the shell. Then, I would use the given location of the center of mass to find the distance of each small element from the axis of rotation (r). However, since the problem is asking for the moment of inertia about a diameter, I would need to take into account the parallel axis theorem, which states that the moment of inertia about an axis parallel to the axis through the center of mass is equal to the moment of inertia about the center of mass plus the product of the mass and the square of the distance between the two axes. I would use this theorem to find the moment of inertia about the desired axis.

In conclusion, to solve these problems, I would use the equation for moment of inertia and take into account the given dimensions and locations of the objects to find the mass and distance of each small element from the axis of rotation. I would also use the parallel axis theorem if