Find out the increase in moment of inertia I of a uniform rod (coefficient of linear expansion α ) about its perpendicular bisector when its temperature is slightly increased by ?T.

This is a short answer type question as classified in NCERT Exemplar

Let the mass and length of a uniform rod be M and l respectively.

Moment of inertia of the rod about its perpendicular bisector I= ml²/12

Increase in length of the rod when temperature is increased by $?$ T is given by

$?l=l\alpha ?T$

So new moment of inertia of the rod = $\frac{M}{12}({l+?l)}^2$ = $\frac{M}{12}\left(l^2+?l^2+2l?l\right)$

As change in length $?l$  is very small therefore neglecting $?l^2$  so we  get

I’= $\frac{M}{12}\left(l^2+2l?l\right)$ = l+mI $\frac{?l}{6}$

Increase in moment of inertia = $\frac{Ml?l}{6}=2\times {\left(\frac{m{l}^2}{12}\right)}^1\frac{?l}{l}=2I\alpha ?T$

<p><span data-teams="true">This is a short answer type question as classified in NCERT Exemplar</span></p><p>Let the mass and length of a uniform rod be M and l respectively.</p><p>Moment of inertia of the rod about its perpendicular bisector I= ml²/12</p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1745477384phpqaBYjq_480x360.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1745477384phpqaBYjq_480x360.jpeg"1 alt="" width="227" height="110"></picture></div></div><p>Increase in length of the rod when temperature is increased by <span contenteditable="false"> <math> <mo>? </mo> </math> </span>T is given by</p><p><span contenteditable="false"> <math> <mo>? </mo> <mi>l</mi> <mo>=</mo> <mi>l</mi> <mi>α</mi> <mo>? </mo> <mi>T</mi> </math> </span></p><p>So new moment of inertia of the rod =<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>M</mi> </mrow> </mrow> <mrow> <mrow> <mn>12</mn> </mrow> </mrow> </mfrac> <mo> (</mo> <msup> <mrow> <mrow> <mi>l</mi> <mo>+</mo> <mo>? </mo> <mi>l</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math> </span> =<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>M</mi> </mrow> </mrow> <mrow> <mrow> <mn>12</mn> </mrow> </mrow> </mfrac> <mfenced separators="|"> <mrow> <mrow> <msup> <mrow> <mrow> <mi>l</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <mo>? </mo> <msup> <mrow> <mrow> <mi>l</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>l</mi> <mo>? </mo> <mi>l</mi> </mrow> </mrow> </mfenced> </math> </span></p><p>As change in length <span contenteditable="false"> <math> <mo>? </mo> <mi>l</mi> </math> </span>&nbsp;is very small therefore neglecting <span contenteditable="false"> <math> <mo>? </mo> <msup> <mrow> <mrow> <mi>l</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math> </span>&nbsp;so we&nbsp; get</p><p>I’=<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>M</mi> </mrow> </mrow> <mrow> <mrow> <mn>12</mn> </mrow> </mrow> </mfrac> <mfenced separators="|"> <mrow> <mrow> <msup> <mrow> <mrow> <mi>l</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>l</mi> <mo>? </mo> <mi>l</mi> </mrow> </mrow> </mfenced> </math> </span> = l+mI<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mo>? </mo> <mi>l</mi> </mrow> </mrow> <mrow> <mrow> <mn>6</mn> </mrow> </mrow> </mfrac> </math> </span></p><p>Increase in moment of inertia =<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>M</mi> <mi>l</mi> <mo>? </mo> <mi>l</mi> </mrow> </mrow> <mrow> <mrow> <mn>6</mn> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> <mo>×</mo> <msup> <mrow> <mrow> <mo> (</mo> <mfrac> <mrow> <mrow> <mi>m</mi> <msup> <mrow> <mrow> <mi>l</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mn>12</mn> </mrow> </mrow> </mfrac> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msup> <mfrac> <mrow> <mrow> <mo>? </mo> <mi>l</mi> </mrow> </mrow> <mrow> <mrow> <mi>l</mi> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> <mi>I</mi> <mi>α</mi> <mo>? </mo> <mi>T</mi> </math> </span></p>

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