The concepts used to solve this problem are law of conservation of angular momentum and the moment of inertia.
First, determine the expressions for the moment of inertia of the disk and the system of the disk and the rod. Finally, calculate the angular frequency of the system of disk and the rod by using the law of conservation of angular momentum.
The angular momentum of the disk is,
Here, is the moment of inertia of the disk and is the angular speed of the disk.
The angular momentum of the system of disk and the rod is,
Here, is the moment of inertia of the system of disk and rod and is the angular speed of the system of disk and rod.
The moment of inertia of a disk of radius about an axis passes through its centre is given by following expression.
The moment of inertia of a rod of length about an axis passes through its centre is given by following expression.
The moment of inertia of the disk is,
Here, is the mass of the disk and is the radius of the disk.
Given that, the length of the rod is equal to the radius of the disk. So, the length of the rod is.
The moment of inertia of the system of disk and rod is,
From the law of conservation of angular momentum, the total initial angular momentum of the system of rod and disk is equal to the total final angular momentum of the system of rod and disk.
Substituteforand for.
Rearrange the above equation for.
Substitute for.
Ans:
The angular frequency of the system of disk and the rod is.
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