Question

One circular rig and one circular disc both are having the same mass and radius. The ratio of their moment of inertia about the axes passing through their centers and perpendicular to their planes, will be \(\\{\mathrm{A}\\} 1: 1\) \(\\{\mathrm{B}\\} 2: 1\) \(\\{C\\} 1: 2\) \(\\{\mathrm{D}\\} 4: 1\)

Short answer

The ratio of the moment of inertia for the circular rig to the circular disc is \(\{B\} 2:1\), as obtained by using the respective moment of inertia formulas \(I_{rig} = M R^2\) and \(I_{disc} = \frac{1}{2} M R^2\), and simplifying the equation for the ratio \(\frac{I_{rig}}{I_{disc}}\).

Step-by-step solution

Determine the moment of inertia formulas for the circular rig and the circular disc

We need to know the formulas for the moment of inertia for both objects. For a circular rig (hollow circle) with mass M and radius R, the moment of inertia is given by: \(I_{rig} = M R^2\). For a circular disc (solid circle) with mass M and radius R, the moment of inertia is given by: \(I_{disc} = \frac{1}{2} M R^2\).

Set up the equation for the ratio of moments of inertia

Now that we have the formulas for the moment of inertia, we need to set up the equation for the ratio of their moments of inertia: \(\frac{I_{rig}}{I_{disc}} = \frac{M R^2}{\frac{1}{2} M R^2}\).

Simplify the equation

Notice that the mass (M) and radius squared (R^2) appear in both the numerator and the denominator. We can simplify the equation by canceling them out: \(\frac{I_{rig}}{I_{disc}} = \frac{M R^2}{\frac{1}{2} M R^2} = \frac{2}{1}\). The ratio of the moment of inertia is 2:1.

Choose the correct answer

Now that we have simplified the equation and found the ratio of the moment of inertia for the circular rig to the circular disc, we can choose the correct answer choice: The correct answer is \(\{B\} 2:1\).