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Chapter 5: Problem 560

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

Expert verified
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

01

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: \(I_{disc} = \frac{1}{2} M R^2\).
02

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: \(\frac{I_{rig}}{I_{disc}} = \frac{M R^2}{\frac{1}{2} M R^2}\).
03

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.
04

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 \(\{B\} 2:1\).

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Most popular questions from this chapter

The moment of inertia of a thin rod of mass \(\mathrm{M}\) and length \(\mathrm{L}\) about an axis passing through the point at a distance $\mathrm{L} / 4$ from one of its ends and perpendicular to the rod is \(\\{\mathrm{A}\\}\left[\left(7 \mathrm{ML}^{2}\right) / 48\right]\) \\{B \\} [ \(\left[\mathrm{ML}^{2} / 12\right]\) \(\\{\mathrm{C}\\}\left[\left(\mathrm{ML}^{2} / 9\right]\right.\) \(\\{\mathrm{D}\\}\left[\left(\mathrm{ML}^{2} / 3\right]\right.\)

A circular disc of radius \(R\) is free to oscillate about an axis passing through a point on its rim and perpendicular to its plane. The disc is turned through an angle of \(60 ?\) and released. Its angular velocity when it reaches the equilibrium position will be \(\\{\mathrm{A}\\} \sqrt{(\mathrm{g} / 3 \mathrm{R})}\) \(\\{\mathrm{B}\\} \sqrt{(2 \mathrm{~g} / 3 \mathrm{R})}\) \(\\{\mathrm{C}\\} \sqrt{(2 \mathrm{~g} / \mathrm{R})}\) \(\\{\mathrm{D}\\} 2 \sqrt{(2 \mathrm{~g} / \mathrm{R})}\)

A uniform rod of length \(\mathrm{L}\) is suspended from one end such that it is free to rotate about an axis passing through that end and perpendicular to the length, what maximum speed must be imparted to the lower end so that the rod completes one full revolution? \(\\{\mathrm{A}\\} \sqrt{(2 \mathrm{~g} \mathrm{~L})}\) \(\\{\mathrm{B}\\} 2 \sqrt{(\mathrm{gL})}\) \(\\{C\\} \sqrt{(6 g L)}\) \(\\{\mathrm{D}\\} 2 \sqrt{(2 g \mathrm{~L})}\)

If the earth were to suddenly contract so that its radius become half of it present radius, without any change in its mass, the duration of the new day will be... \(\\{\mathrm{A}\\} 6 \mathrm{hr}\) \\{B \(12 \mathrm{hr}\) \(\\{\mathrm{C}\\} 18 \mathrm{hr}\) \\{D \(\\} 30 \mathrm{hr}\)

Moment of inertia of a sphere of mass \(\mathrm{M}\) and radius \(\mathrm{R}\) is \(\mathrm{I}\). keeping mass constant if graph is plotted between \(\mathrm{I}\) and \(\mathrm{R}\) then its form would be.
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