365870
Assertion : Two circular discs of equal mass and thickness made of different materials, will have same moment of inertia about their central axes of rotation. Reason : Moment of inertia depends upon the distribution of mass in the body
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Both Assertion and reason are incorrect.
Explanation:
The moment of inertia is given by \(I=\sum m r^{2}\). The moment of inertia depends on the mass and distribution of mass w.r.t the axis of rotation in the body, so that moment of inertia of two circular discs having equal mass and thickness made of different material will be same.
AIIMS - 2013
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365871
Assertion : Two circular discs of equal radius, mass and volume made of different materials. will have same moment of inertia about their central axes of rotation. Reason : Moment of inertia depends upon the mass of the body only.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
In assertion, density does not play role as \(M, V\) are same. While the assertion is true that two circular discs of equal masses radius and thickness made of different materials will have the same moment of inertia (I) about their central axes of rotation, the reason provided is incorrect.Moment of inertia depends on both mass \((M)\) and the distribution of mass around the axis of rotation, (indicated by radius of gyration \(K\) )not just the mass itself \(I=M K^{2}\). So correct option is (3).
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365872
The ratio of the radii of gyration of a circular disc to that of a circular ring, each of same mass and radius, around their respective axes is
1 \(\sqrt 2 \,\,:\,\,\,\,1\)
2 \(\sqrt 3 \,\,:\sqrt 2 \)
3 \(\sqrt 2 :\sqrt 3 \)
4 \(\,\,\,1\,\,\,\,:\sqrt 2 \)
Explanation:
As we know that radius of gyration \(k=\sqrt{\dfrac{I}{m}}\) So, for two different cases \(\frac{{{k_{{\rm{ring }}}}}}{{{k_{{\rm{disc }}}}}} = \sqrt {\frac{{{I_{{\rm{ring }}}}}}{{{I_{{\rm{disc }}}}}}} = \sqrt {\frac{{M{R^2}}}{{\frac{1}{2}M{R^2}}}} \) \(\frac{{{k_{{\rm{disc }}}}}}{{{k_{{\rm{ring }}}}}} = \frac{1}{{\sqrt 2 }}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365873
Two discs having mass ratio \(1: 2\) and diameter ratio \(2: 1\), then find the ratio of moment of inertia.
1 \(2: 1\)
2 \(1: 1\)
3 \(1: 2\)
4 \(2: 3\)
Explanation:
Given, mass ratio of two discs, \({m_1}:{m_2} = 1:2,{\text{ }}i.e.{\text{ }}\frac{{{m_1}}}{{{m_2}}} = \frac{1}{2}\) and diameter ratio \(\dfrac{d_{1}}{d_{2}}=\dfrac{2}{1} \Rightarrow \dfrac{r_{1}}{r_{2}}=\dfrac{2}{1}\) \(\therefore\) Ratio of their moment of inertia, \(\begin{aligned}& \dfrac{I_{1}}{I_{2}}=\dfrac{\dfrac{m_{1} r_{1}^{2}}{2}}{\dfrac{m_{2} r_{2}^{2}}{2}}=\dfrac{m_{1}}{m_{2}} \cdot\left(\dfrac{r_{1}}{r_{2}}\right)^{2}=\dfrac{1}{2}\left(\dfrac{2}{1}\right)^{2}=\dfrac{2}{1} \\& \therefore \quad I_{1}: I_{2}=2: 1\end{aligned}\)
365870
Assertion : Two circular discs of equal mass and thickness made of different materials, will have same moment of inertia about their central axes of rotation. Reason : Moment of inertia depends upon the distribution of mass in the body
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Both Assertion and reason are incorrect.
Explanation:
The moment of inertia is given by \(I=\sum m r^{2}\). The moment of inertia depends on the mass and distribution of mass w.r.t the axis of rotation in the body, so that moment of inertia of two circular discs having equal mass and thickness made of different material will be same.
AIIMS - 2013
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365871
Assertion : Two circular discs of equal radius, mass and volume made of different materials. will have same moment of inertia about their central axes of rotation. Reason : Moment of inertia depends upon the mass of the body only.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
In assertion, density does not play role as \(M, V\) are same. While the assertion is true that two circular discs of equal masses radius and thickness made of different materials will have the same moment of inertia (I) about their central axes of rotation, the reason provided is incorrect.Moment of inertia depends on both mass \((M)\) and the distribution of mass around the axis of rotation, (indicated by radius of gyration \(K\) )not just the mass itself \(I=M K^{2}\). So correct option is (3).
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365872
The ratio of the radii of gyration of a circular disc to that of a circular ring, each of same mass and radius, around their respective axes is
1 \(\sqrt 2 \,\,:\,\,\,\,1\)
2 \(\sqrt 3 \,\,:\sqrt 2 \)
3 \(\sqrt 2 :\sqrt 3 \)
4 \(\,\,\,1\,\,\,\,:\sqrt 2 \)
Explanation:
As we know that radius of gyration \(k=\sqrt{\dfrac{I}{m}}\) So, for two different cases \(\frac{{{k_{{\rm{ring }}}}}}{{{k_{{\rm{disc }}}}}} = \sqrt {\frac{{{I_{{\rm{ring }}}}}}{{{I_{{\rm{disc }}}}}}} = \sqrt {\frac{{M{R^2}}}{{\frac{1}{2}M{R^2}}}} \) \(\frac{{{k_{{\rm{disc }}}}}}{{{k_{{\rm{ring }}}}}} = \frac{1}{{\sqrt 2 }}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365873
Two discs having mass ratio \(1: 2\) and diameter ratio \(2: 1\), then find the ratio of moment of inertia.
1 \(2: 1\)
2 \(1: 1\)
3 \(1: 2\)
4 \(2: 3\)
Explanation:
Given, mass ratio of two discs, \({m_1}:{m_2} = 1:2,{\text{ }}i.e.{\text{ }}\frac{{{m_1}}}{{{m_2}}} = \frac{1}{2}\) and diameter ratio \(\dfrac{d_{1}}{d_{2}}=\dfrac{2}{1} \Rightarrow \dfrac{r_{1}}{r_{2}}=\dfrac{2}{1}\) \(\therefore\) Ratio of their moment of inertia, \(\begin{aligned}& \dfrac{I_{1}}{I_{2}}=\dfrac{\dfrac{m_{1} r_{1}^{2}}{2}}{\dfrac{m_{2} r_{2}^{2}}{2}}=\dfrac{m_{1}}{m_{2}} \cdot\left(\dfrac{r_{1}}{r_{2}}\right)^{2}=\dfrac{1}{2}\left(\dfrac{2}{1}\right)^{2}=\dfrac{2}{1} \\& \therefore \quad I_{1}: I_{2}=2: 1\end{aligned}\)
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PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365870
Assertion : Two circular discs of equal mass and thickness made of different materials, will have same moment of inertia about their central axes of rotation. Reason : Moment of inertia depends upon the distribution of mass in the body
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Both Assertion and reason are incorrect.
Explanation:
The moment of inertia is given by \(I=\sum m r^{2}\). The moment of inertia depends on the mass and distribution of mass w.r.t the axis of rotation in the body, so that moment of inertia of two circular discs having equal mass and thickness made of different material will be same.
AIIMS - 2013
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365871
Assertion : Two circular discs of equal radius, mass and volume made of different materials. will have same moment of inertia about their central axes of rotation. Reason : Moment of inertia depends upon the mass of the body only.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
In assertion, density does not play role as \(M, V\) are same. While the assertion is true that two circular discs of equal masses radius and thickness made of different materials will have the same moment of inertia (I) about their central axes of rotation, the reason provided is incorrect.Moment of inertia depends on both mass \((M)\) and the distribution of mass around the axis of rotation, (indicated by radius of gyration \(K\) )not just the mass itself \(I=M K^{2}\). So correct option is (3).
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365872
The ratio of the radii of gyration of a circular disc to that of a circular ring, each of same mass and radius, around their respective axes is
1 \(\sqrt 2 \,\,:\,\,\,\,1\)
2 \(\sqrt 3 \,\,:\sqrt 2 \)
3 \(\sqrt 2 :\sqrt 3 \)
4 \(\,\,\,1\,\,\,\,:\sqrt 2 \)
Explanation:
As we know that radius of gyration \(k=\sqrt{\dfrac{I}{m}}\) So, for two different cases \(\frac{{{k_{{\rm{ring }}}}}}{{{k_{{\rm{disc }}}}}} = \sqrt {\frac{{{I_{{\rm{ring }}}}}}{{{I_{{\rm{disc }}}}}}} = \sqrt {\frac{{M{R^2}}}{{\frac{1}{2}M{R^2}}}} \) \(\frac{{{k_{{\rm{disc }}}}}}{{{k_{{\rm{ring }}}}}} = \frac{1}{{\sqrt 2 }}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365873
Two discs having mass ratio \(1: 2\) and diameter ratio \(2: 1\), then find the ratio of moment of inertia.
1 \(2: 1\)
2 \(1: 1\)
3 \(1: 2\)
4 \(2: 3\)
Explanation:
Given, mass ratio of two discs, \({m_1}:{m_2} = 1:2,{\text{ }}i.e.{\text{ }}\frac{{{m_1}}}{{{m_2}}} = \frac{1}{2}\) and diameter ratio \(\dfrac{d_{1}}{d_{2}}=\dfrac{2}{1} \Rightarrow \dfrac{r_{1}}{r_{2}}=\dfrac{2}{1}\) \(\therefore\) Ratio of their moment of inertia, \(\begin{aligned}& \dfrac{I_{1}}{I_{2}}=\dfrac{\dfrac{m_{1} r_{1}^{2}}{2}}{\dfrac{m_{2} r_{2}^{2}}{2}}=\dfrac{m_{1}}{m_{2}} \cdot\left(\dfrac{r_{1}}{r_{2}}\right)^{2}=\dfrac{1}{2}\left(\dfrac{2}{1}\right)^{2}=\dfrac{2}{1} \\& \therefore \quad I_{1}: I_{2}=2: 1\end{aligned}\)
365870
Assertion : Two circular discs of equal mass and thickness made of different materials, will have same moment of inertia about their central axes of rotation. Reason : Moment of inertia depends upon the distribution of mass in the body
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Both Assertion and reason are incorrect.
Explanation:
The moment of inertia is given by \(I=\sum m r^{2}\). The moment of inertia depends on the mass and distribution of mass w.r.t the axis of rotation in the body, so that moment of inertia of two circular discs having equal mass and thickness made of different material will be same.
AIIMS - 2013
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365871
Assertion : Two circular discs of equal radius, mass and volume made of different materials. will have same moment of inertia about their central axes of rotation. Reason : Moment of inertia depends upon the mass of the body only.
1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.
2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.
3 Assertion is correct but Reason is incorrect.
4 Assertion is incorrect but reason is correct.
Explanation:
In assertion, density does not play role as \(M, V\) are same. While the assertion is true that two circular discs of equal masses radius and thickness made of different materials will have the same moment of inertia (I) about their central axes of rotation, the reason provided is incorrect.Moment of inertia depends on both mass \((M)\) and the distribution of mass around the axis of rotation, (indicated by radius of gyration \(K\) )not just the mass itself \(I=M K^{2}\). So correct option is (3).
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365872
The ratio of the radii of gyration of a circular disc to that of a circular ring, each of same mass and radius, around their respective axes is
1 \(\sqrt 2 \,\,:\,\,\,\,1\)
2 \(\sqrt 3 \,\,:\sqrt 2 \)
3 \(\sqrt 2 :\sqrt 3 \)
4 \(\,\,\,1\,\,\,\,:\sqrt 2 \)
Explanation:
As we know that radius of gyration \(k=\sqrt{\dfrac{I}{m}}\) So, for two different cases \(\frac{{{k_{{\rm{ring }}}}}}{{{k_{{\rm{disc }}}}}} = \sqrt {\frac{{{I_{{\rm{ring }}}}}}{{{I_{{\rm{disc }}}}}}} = \sqrt {\frac{{M{R^2}}}{{\frac{1}{2}M{R^2}}}} \) \(\frac{{{k_{{\rm{disc }}}}}}{{{k_{{\rm{ring }}}}}} = \frac{1}{{\sqrt 2 }}\)
PHXI07:SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
365873
Two discs having mass ratio \(1: 2\) and diameter ratio \(2: 1\), then find the ratio of moment of inertia.
1 \(2: 1\)
2 \(1: 1\)
3 \(1: 2\)
4 \(2: 3\)
Explanation:
Given, mass ratio of two discs, \({m_1}:{m_2} = 1:2,{\text{ }}i.e.{\text{ }}\frac{{{m_1}}}{{{m_2}}} = \frac{1}{2}\) and diameter ratio \(\dfrac{d_{1}}{d_{2}}=\dfrac{2}{1} \Rightarrow \dfrac{r_{1}}{r_{2}}=\dfrac{2}{1}\) \(\therefore\) Ratio of their moment of inertia, \(\begin{aligned}& \dfrac{I_{1}}{I_{2}}=\dfrac{\dfrac{m_{1} r_{1}^{2}}{2}}{\dfrac{m_{2} r_{2}^{2}}{2}}=\dfrac{m_{1}}{m_{2}} \cdot\left(\dfrac{r_{1}}{r_{2}}\right)^{2}=\dfrac{1}{2}\left(\dfrac{2}{1}\right)^{2}=\dfrac{2}{1} \\& \therefore \quad I_{1}: I_{2}=2: 1\end{aligned}\)