Moment of Inertia Calculation for Rotating Disc

[bidhati]

Homework Statement

A horizontal disc of diameter 12.0 cm is spinning freely about a vertical axis
through its centre at an angular speed of 72 revolutions per minute. A piece
of putty of mass 5.0 g drops on to and sticks to the disc a distance of 4.0 cm
from the centre. The angular speed reduces to 60 revolutions per minute.
Calculate the moment of inertia of the disc. You should assume that no
external torques are applied to the system during this process.

Homework Equations

conservation of momentum
I0w0=Ifwf

parallel axis theorim
Iw=Iw+mr^2

The Attempt at a Solution

combining two equations gives Iw=Iw +mr^2
but do I need to convert angular speed to radians per sec? and I assume the mass is the putty mass not the disc?

 

[Mentz114]

If you knew the mass of the disc, the question would have no point would it, because that's what you're asked to find ?

Yes, use radians/sec for angular velocity.

 

[m2003]

Yes, you will need to convert the angular speed to radians per second in order to use the equations correctly. The mass in this case would be the total mass of the disc and the putty, as they are now considered one system. So, the equation would be I0w0=Ifwf, where I0 is the initial moment of inertia of the disc, w0 is the initial angular speed of the disc, If is the final moment of inertia of the disc with the added putty, and wf is the final angular speed of the disc.

To calculate the moment of inertia of the disc, you can use the parallel axis theorem, which states that the moment of inertia of a body is equal to the sum of its mass multiplied by the square of the distance between the axis of rotation and the center of mass, plus the moment of inertia of the body when it rotates about its center of mass. In this case, the distance between the axis of rotation and the center of mass is given as 4.0 cm. So, the equation would be I=Icm + mr^2, where Icm is the moment of inertia of the disc when it rotates about its center of mass.

To calculate the final moment of inertia, you can rearrange the equation I0w0=Ifwf to solve for If. Then, plug in the known values and solve for If. Once you have the final moment of inertia, you can use the parallel axis theorem to calculate the moment of inertia of the disc when it rotates about its center of mass. Finally, you can add this value to the moment of inertia of the added putty (which is just mr^2) to get the total moment of inertia of the disc with the added putty.