A ball is released from rest point P of a smooth semi-spherical vessel as shown is figure. The ratio of the centripetal force and normal reaction on the ball at point Q is A while angular position of point Q is a with respect to point P. Which of the following graphs represent the correct relation between A and a when ball goes from Q to R?      

 

 

 

 

Correct Option - 2


Detailed Solution:

NLM at point Q

N – mg sin a =  $\frac{m{v}^2}{R}$  

$N=\frac{m{v}^2}{R}+mgsin\alpha$           - (1)

COE : between P & Q

$\frac{m{v}^2}{R}=2mgsin\alpha$               - (2)

Centripetal force = 2mg sin a

 N (Normal Reaction) = 2mg sin a + mg sin a

  = 3mg sin a

$\frac{Centripetalforce}{NormalReaction}=\frac{2mgsin\alpha }{3mgsin\alpha }=\frac{2}{3}$             

 

<p>NLM at point Q</p><p>N – mg sin a =&nbsp;<span contenteditable="false"> <math> <mrow> <mfrac> <mrow> <mi>m</mi> <msup> <mrow> <mi>v</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>R</mi> </mrow> </mfrac> </mrow> </math> </span>&nbsp;</p><p><span contenteditable="false"> <math> <mrow> <mi>N</mi> <mo>=</mo> <mfrac> <mrow> <mi>m</mi> <msup> <mrow> <mi>v</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>R</mi> </mrow> </mfrac> <mo>+</mo> <mi>m</mi> <mi>g</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>α</mi> </mrow> </math> </span>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; - (1)</p><p>COE : between P &amp; Q</p><p><span contenteditable="false"> <math> <mrow> <mfrac> <mrow> <mi>m</mi> <msup> <mrow> <mi>v</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>R</mi> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> <mi>m</mi> <mi>g</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>α</mi> </mrow> </math> </span>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; - (2)</p><p>Centripetal force = 2mg sin a</p><p>&nbsp;N (Normal Reaction) = 2mg sin a + mg sin a</p><p>&nbsp; = 3mg sin a</p><p><span contenteditable="false"> <math> <mrow> <mfrac> <mrow> <mi>C</mi> <mi>e</mi> <mi>n</mi> <mi>t</mi> <mi>r</mi> <mi>i</mi> <mi>p</mi> <mi>e</mi> <mi>t</mi> <mi>a</mi> <mi>l</mi> <mtext> </mtext> <mtext> </mtext> <mi>f</mi> <mi>o</mi> <mi>r</mi> <mi>c</mi> <mi>e</mi> </mrow> <mrow> <mi>N</mi> <mi>o</mi> <mi>r</mi> <mi>m</mi> <mi>a</mi> <mi>l</mi> <mtext> </mtext> <mtext> </mtext> <mi>R</mi> <mi>e</mi> <mi>a</mi> <mi>c</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> <mi>n</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> <mi>g</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>α</mi> </mrow> <mrow> <mn>3</mn> <mi>m</mi> <mi>g</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>α</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> </mrow> </math> </span> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;</p><div><div><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1758002518php4g9tLF.jpeg" alt="" width="203" height="329" loading="lazy"></picture></div></div><p>&nbsp;</p>

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