A rod of mass M and length L is lying on a horizontal frictionless surface. A particle of mass ‘m’ travelling along the surface hits at one end of the rod with a velocity ‘u’ in a direction perpendicular to the rod. The collision is completely elastic. After collision, particle comes to rest. The ratio of masses $\left(\frac{m}{M}\right)is\frac{1}{x}.$  The value of ‘x’ will be__________.

Using conservation of linear momentum, we can write

 

$mu=M{v}_1\Rightarrow v_1=\frac{mu}{M}........\left(1\right)$           

Using conservation of angular momentum about centre of mass of Rod, we can write

$v_1+\frac{\mathcal{l}\omega }{2}=u\Rightarrow \frac{mu}{M}+\frac{3mu}{M}=u\Rightarrow \frac{m}{M}=\frac{1}{4}$     . (2)

Using definition of e, we can write

$v_1+\frac{\mathcal{l}\omega }{2}=u\Rightarrow \frac{mu}{M}+\frac{3mu}{M}=u\Rightarrow \frac{m}{M}=\frac{1}{4}$  

<p>Using conservation of linear momentum, we can write</p><p>&nbsp;</p><div><div><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1754569488php8rpDyC.jpeg" alt="" width="610" height="142" loading="lazy"></picture></div></div><p><span contenteditable="false"> <math> <mrow> <mi>m</mi> <mi>u</mi> <mo>=</mo> <mi>M</mi> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>⇒</mo> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>m</mi> <mi>u</mi> </mrow> <mrow> <mi>M</mi> </mrow> </mfrac> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo> (</mo> <mrow> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </math> </span> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1816094890"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;</p><p>Using conservation of angular momentum about centre of mass of Rod, we can write</p><p><br><span contenteditable="false"> <math> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <mi mathvariant="script">l</mi> <mi>ω</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>=</mo> <mi>u</mi> <mo>⇒</mo> <mfrac> <mrow> <mi>m</mi> <mi>u</mi> </mrow> <mrow> <mi>M</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>3</mn> <mi>m</mi> <mi>u</mi> </mrow> <mrow> <mi>M</mi> </mrow> </mfrac> <mo>=</mo> <mi>u</mi> <mo>⇒</mo> <mfrac> <mrow> <mi>m</mi> </mrow> <mrow> <mi>M</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> </mrow> </math> </span> &nbsp; &nbsp;&lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1816094891"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;. (2)</p><p>Using definition of e, we can write</p><p><span contenteditable="false"> <math> <mrow> <msub> <mrow> <mi>v</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <mi mathvariant="script">l</mi> <mi>ω</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>=</mo> <mi>u</mi> <mo>⇒</mo> <mfrac> <mrow> <mi>m</mi> <mi>u</mi> </mrow> <mrow> <mi>M</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>3</mn> <mi>m</mi> <mi>u</mi> </mrow> <mrow> <mi>M</mi> </mrow> </mfrac> <mo>=</mo> <mi>u</mi> <mo>⇒</mo> <mfrac> <mrow> <mi>m</mi> </mrow> <mrow> <mi>M</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> </mrow> </math> </span>&nbsp;&lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1816094892"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;</p>

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