The ratio of radius of gyration of a solid sphere of mass <math> <mi>M</mi> </math> and radius <math> <mi mathvariant="normal">R</mi> </math> about its own axis to the radius of gyration of the thin hollow sphere of same mass and radius about its axis is:

Let the gravitational field is zero at a distance x from the mass m . G m x 2 = G 9 m ( R - x ) 2  ⇒ R - x = 3 x or x = R 4 Gravitational potential at <!-- [if gte msEquation 12]>R4

The ratio of radius of gyration of a solid sphere of mass <math> <mi>M</mi> </math> and radius <math> <mi mathvariant="normal">R</mi> </math> &nbsp;about its own axis to the radius of gyration of the thin hollow sphere of same mass and radius about its axis is:

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Physics Class 11th Physics System of Particles and Rotational Motion

The ratio of radius of gyration of a solid sphere of mass $M$ and radius $R$ about its own axis to the radius of gyration of the thin hollow sphere of same mass and radius about its axis is:

Option 1 -

$5 : 2$

Option 2 -

$3 : 5$

Option 3 -

$5 : 3$

Option 4 -

$2 : 5$

3 Views | Posted a month ago

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1 Answer

Answered by

alok kumar singh | Contributor-Level 10

a month ago

Correct Option - 4

Detailed Solution:

Let the gravitational field is zero at a distance $x$ from the mass $m$ .

$\frac{G m}{x^{2}} = \frac{G 9 m}{(R - x)^{2}}$

$\Rightarrow R - x = 3 x$ or $x = \frac{R}{4}$

Gravitational potential at $\frac{R}{4}$

$= - \frac{G m}{\frac{R}{4}} - \frac{G 9 m}{\frac{3 R}{4}}$

$= - \frac{4 G m}{R} - \frac{12 G m}{R}$

$= - \frac{16 G m}{R}$

<div><div><picture><source srcset="" media="(max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1756896174php9pQBGk.jpeg" alt="" width="412" height="150" loading="lazy"></picture></div><div>Let the gravitational field is zero at a distance <math> <mi mathvariant="normal">x</mi> </math> from the mass <math> <mi mathvariant="normal">m</mi> </math> . <math> <mfrac> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mi mathvariant="normal">x</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mn>9</mn> <mtext> </mtext> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mi mathvariant="normal">R</mi> <mo>-</mo> <mi mathvariant="normal">x</mi> <msup> <mrow> <mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> </mfrac> </math>  <math> <mo>⇒</mo> <mi mathvariant="normal">R</mi> <mo>-</mo> <mi mathvariant="normal">x</mi> <mo>=</mo> <mn>3</mn> <mi mathvariant="normal">x</mi> </math> or <math> <mi mathvariant="normal">x</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> </mrow> </mrow> </mfrac> </math> Gravitational potential at <img src="file:///C:/Users/ALOK~1.SIN/AppData/Local/Temp/msohtmlclip1/01/clip_image012.png" width="7" height="25"> <math> <mfrac> <mrow> <mrow> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> </mrow> </mrow> </mfrac> </math>  <math> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfrac> <mo>-</mo> <mfrac> <mrow> <mrow> <mi mathvariant="normal">G</mi> <mn>9</mn> <mtext> </mtext> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn>3</mn> <mi mathvariant="normal">R</mi> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfrac> </math>  <math> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mrow> <mn>4</mn> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="normal">R</mi> </mrow> </mrow> </mfrac> <mo>-</mo> <mfrac> <mrow> <mrow> <mn>12</mn> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="normal">R</mi> </mrow> </mrow> </mfrac> </math>  <math> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mrow> <mn>16</mn> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="normal">R</mi> </mrow> </mrow> </mfrac> </math> </div></div>

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The ratio of radius of gyration of a solid sphere of mass $M$ and radius $R$ about its own axis to the radius of gyration of the thin hollow sphere of same mass and radius about its axis is:

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The ratio of radius of gyration of a solid sphere of mass $M$ and radius $R$ about its own axis to the radius of gyration of the thin hollow sphere of same mass and radius about its axis is: