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Chapter 5: Problem 566

Two disc of same thickness but of different radii are made of two different materials such that their masses are same. The densities of the materials are in the ratio \(1: 3\). The moment of inertia of these disc about the respective axes passing through their centres and perpendicular to their planes will be in the ratio. \(\\{\mathrm{A}\\} 1: 3\) \\{B\\} \(3: 1\) \\{C\\} \(1: 9\) \(\\{\mathrm{D}\\} 9: 1\)

Short Answer

Expert verified
The ratio of the moments of inertia of the two discs is \({B}\\ \, 3:1 \).

Step by step solution

01

Determine the formula for the moment of inertia of a disc

The moment of inertia of a disc about an axis passing through its center and perpendicular to its plane is given by the formula: \[I = \frac{1}{2}MR^2\] Where: \(I\) is the moment of inertia \(M\) is the mass of the disc \(R\) is the radius of the disc
02

Set up the relationship between the given densities and radii

The density \(\rho\) of a material is defined as the mass per unit volume: \(\rho = \frac{M}{V}\). For a circular disc, the volume can be expressed in terms of the radius and thickness: \(V = \pi R^2 h\), where \(h\) is the thickness. Since both discs have the same mass, we can write: \[\frac{\rho_1}{\rho_2} = \frac{3}{1} = \frac{M_1 R_1^2 h}{M_2 R_2^2 h}\] Since \(M_1 = M_2\) and the thicknesses are equal, we can cancel them out: \[\frac{3}{1} = \frac{R_1^2}{R_2^2}\]
03

Find the ratio of the moments of inertia of the two discs

We want to find the ratio \(\frac{I_1}{I_2}\), where \(I_1 = \frac{1}{2}M_1R_1^2\) and \(I_2 = \frac{1}{2}M_2R_2^2\). Using the known ratio of densities, we can calculate the ratio of moments of inertia: \[\frac{I_1}{I_2} = \frac{\frac{1}{2}M_1R_1^2}{\frac{1}{2}M_2R_2^2} = \frac{R_1^2}{R_2^2}\] Using the relation obtained in Step 2 for \(\frac{R_1^2}{R_2^2}\), we can write: \[\frac{I_1}{I_2} = \frac{3}{1} = 3:1\] So the answer to the problem is: \[{B}\\ \, 3:1 \]

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Most popular questions from this chapter

Statement \(-1\) - Two cylinder one hollow and other solid (wood) with the same mass and identical dimensions are simultaneously allowed to roll without slipping down an inclined plane from the same height. The hollow will reach the bottom of inclined plane first. Statement \(-2-\mathrm{By}\) the principle of conservation of energy, the total kinetic energies of both the cylinders are identical when they reach the bottom of the incline. \(\\{\mathrm{A}\\}\) Statement \(-1\) is correct (true), Statement \(-2\) is true and Statement- 2 is correct explanation for Statement \(-1\) \\{B \\} Statement \(-1\) is true, statement \(-2\) is true but statement- 2 is not the correct explanation four statement \(-1\). \\{C\\} Statement - 1 is true, statement- 2 is false \\{D \(\\}\) Statement- 2 is false, statement \(-2\) is true

A particle performing uniform circular motion has angular momentum \(L\)., its angular frequency is doubled and its \(K . E\). halved, then the new angular momentum is \(\\{\mathrm{A}\\} 1 / 2\) \\{B \(\\} 1 / 4\) \(\\{\mathrm{C}\\} 2 \mathrm{~L}\) \(\\{\mathrm{D}\\} 4 \mathrm{~L}\)

The M.I of a disc of mass \(\mathrm{M}\) and radius \(\mathrm{R}\) about an axis passing through the centre \(\mathrm{O}\) and perpendicular to the plane of disc is \(\left(\mathrm{MR}^{2} / 2\right)\). If one quarter of the disc is removed the new moment of inertia of disc will be..... \(\\{\mathrm{A}\\}\left(\mathrm{MR}^{2} / 3\right)\) \(\\{B\\}\left(M R^{2} / 4\right)\) \(\\{\mathrm{C}\\}(3 / 8) \mathrm{MR}^{2}\) \(\\{\mathrm{D}\\}(3 / 2) \mathrm{MR}^{2}\)

If the earth is treated as a sphere of radius \(R\) and mass \(M\). Its angular momentum about the axis of rotation with period \(\mathrm{T}\) is..... \(\\{\mathrm{A}\\}\left(\pi \mathrm{MR}^{3} / \mathrm{T}\right)\) \(\\{\mathrm{B}\\}\left(\operatorname{MR}^{2} \pi / \mathrm{T}\right)\) \(\\{C\\}\left(2 \pi \mathrm{MR}^{2} / 5 \mathrm{~T}\right)\) \(\\{\mathrm{D}\\}\left(4 \pi \mathrm{MR}^{2} / 5 \mathrm{~T}\right)\)

A wheel rotates with a constant acceleration of $2.0 \mathrm{rad} / \mathrm{sec}^{2}$ If the wheel start from rest. The number of revolution it makes in the first ten seconds will be approximately. \(\\{\mathrm{A}\\} 8\) \\{B \\} 16 \(\\{\mathrm{C}\\} 24\) \(\\{\mathrm{D}\\} 32\)
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