Moment of inertia with varying density

In summary, the conversation was about finding the moment of inertia of a sphere with varying density, using equations from a website and substituting the given density. The final answer was 2/15mr^2, but the other person pointed out that it should be between the solid sphere and thin shell values, and suggested it should be 1/2Mr^2. They also asked for more working to check if the constant c was correct, and suggested their own answer of 4/9MR^2.
  • #1
PhilosophyofPhysics
95
1
I had to find the moment of inertia of a sphere of varying density where the density is some constant times the radius.

I used the equations from here

http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html

I inserted this d=cr in for the density.

For the mass I used dm=4(pi)(d)r^2dr, inserted the density, integrated and solved for c.

After substitution I the answer I got was 2/15mr^2.

Does this sound right?
 
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  • #2
The answer should be between the solid sphere value and the thin shell value, so your answer is too small. I did a rough calculation and got 1/2Mr^2.

Can you show more of your working ?
 
  • #3
your dm element seems ok if you are taking thin shell as your differential element. I got [tex]\frac{4}{9} MR^2[/tex]. you should check whether you have got constant c correct.
 

FAQ: Moment of inertia with varying density

What is moment of inertia with varying density?

Moment of inertia with varying density is a measure of an object's resistance to rotational motion when its density is not uniform throughout its mass. It takes into account the distribution of mass within an object and how it affects its rotational inertia.

How is moment of inertia with varying density calculated?

The moment of inertia with varying density is calculated by dividing the integral of the product of the density and the square of the distance from the axis of rotation by the total mass of the object. This can also be represented as the sum of all the individual moments of inertia of infinitesimal elements that make up the object.

What are the applications of moment of inertia with varying density?

Moment of inertia with varying density is used in various fields such as physics, engineering, and materials science. It is important in understanding the rotational behavior of objects with non-uniform mass distribution, such as a spinning top or a rotating satellite.

How does moment of inertia with varying density differ from moment of inertia with constant density?

The moment of inertia with constant density assumes that the mass of an object is evenly distributed throughout its volume. On the other hand, the moment of inertia with varying density takes into account the varying distribution of mass within an object, resulting in a different value for the moment of inertia.

How can moment of inertia with varying density be experimentally determined?

Moment of inertia with varying density can be determined experimentally by measuring the angular acceleration of an object of known mass and shape when subjected to a known torque. The experimental value can then be compared to the theoretical value calculated using the above mentioned formula.

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