HELP - ROTATIONAL INERTIA (no numbers given)

[Quarkn]

Homework Statement

A bowling ball made for a child has half the radius of an adult bowling ball. They are made of the same material (and therefore have the same mass per unit volume). By what factor is a) mass and b) rotational inertia if the child's ball reduced compared with the adult ball?

Homework Equations

I=MR²
(sphere) I=(2/5)MR²

The Attempt at a Solution

I only got to: R(adult) = (1/2)R (child).

PS. the answers to a) reduced by a factor of 8 and b)reduced by a factor of 32

 

[physicsvalk]

Set the two densities equal to each other, and then examine how the masses vary (since the radius of the child's bowling ball differs from the adult's). Based on this, you should have enough information to determine its moment of inertia.

 

[Quarkn]

Set the two densities equal to each other, and then examine how the masses vary (since the radius of the child's bowling ball differs from the adult's). Based on this, you should have enough information to determine its moment of inertia.

So would you find the volume of each spheres first?

And if i set the two densities equal, wouldn't the masses just cancel?

ex. D=M/V, M/v=M/V ?

 

[physicsvalk]

Yes, you would need to find the volumes.

The densities are said to be equal and the volumes differ, therefore, the masses can't be the same.

 

[asteriatic]

I would first like to clarify that rotational inertia, also known as moment of inertia, is a measure of an object's resistance to changes in its rotational motion. It is dependent on the mass, shape, and distribution of the object's mass.

In this scenario, we are comparing the rotational inertia of an adult bowling ball to that of a child's bowling ball. We are given that the child's ball has half the radius of the adult ball and that they are made of the same material with the same mass per unit volume.

To calculate the rotational inertia, we can use the equation I=MR² for a solid sphere. However, since the balls are not solid spheres, we need to use the equation for a hollow sphere, which is I=(2/3)MR².

a) For the mass, we can assume that the density of both balls is the same, as they are made of the same material. This means that the mass will be directly proportional to the volume, which is proportional to the radius cubed. Therefore, the mass of the child's ball will be (1/8) of the mass of the adult ball.

b) For the rotational inertia, we can use the equation I=(2/3)MR². As the mass is reduced by a factor of 1/8, the rotational inertia will be reduced by the same factor. Therefore, the child's ball will have a rotational inertia that is (1/8) of the adult ball's rotational inertia.

In conclusion, the mass of the child's ball will be reduced by a factor of 8, while the rotational inertia will be reduced by a factor of 8 as well.