A block of mass m slides on the wooden wedge, which in turn slides backward on the horizontal surface. The acceleration of the block with respect to the wedge is: Given m = 8 kg, M = 16kg. Assume all the surfaces shown in the figure to be frictionless.

 

Correct Option - 3


Detailed Solution:

Suppose acceleration of wedge is a and acceleration of block w. r.t. wedge is a₁ then N cos60° = Ma = 16 a -> N = 32 a

For block w.r.t. wedge

N + 8a sin 30° = 8g cos 30°

N = 8g cos 30° - 8a sin 30°

->32a = 8g cos 30° - 8a sin 30°

->a = $\frac{\sqrt[]{3}}{9}g$  

Now for 8 kg,

$8gsin30°+8acos30°=m{a}_1$  

$a_1=g\times \frac{1}{2}+\frac{\sqrt[]{3}}{9}g\times \frac{\sqrt[]{3}}{2}$  

$=\frac{2}{3}g$  

 

<p>Suppose acceleration of wedge is a and acceleration of block w. r.t. wedge is a₁ then N cos60° = Ma = 16 a -&gt; N = 32 a</p><p>For block w.r.t. wedge</p><p>N + 8a sin 30° = 8g cos 30°</p><p>N = 8g cos 30° - 8a sin 30°</p><p>-&gt;32a = 8g cos 30° - 8a sin 30°</p><p>-&gt;a =<span contenteditable="false"> <math> <mrow> <mfrac> <mrow> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> </mrow> <mrow> <mn>9</mn> </mrow> </mfrac> <mi>g</mi> </mrow> </math> </span>&nbsp;&lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1815998163"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;</p><p>Now for 8 kg, </p><p><span contenteditable="false"> <math> <mrow> <mn>8</mn> <mi>g</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mn>3</mn> <mn>0</mn> <mo>°</mo> <mo>+</mo> <mn>8</mn> <mi>a</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mn>3</mn> <mn>0</mn> <mo>°</mo> <mo>=</mo> <mi>m</mi> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </math> </span>&nbsp;&lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1815998164"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;</p><p><span contenteditable="false"> <math> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>g</mi> <mo>×</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> </mrow> <mrow> <mn>9</mn> </mrow> </mfrac> <mi>g</mi> <mo>×</mo> <mfrac> <mrow> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> </mrow> <mrow> <mn>2</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="_1815998165"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;</p><p><br><span contenteditable="false"> <math> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mi>g</mi> </mrow> </math> </span> &nbsp;&lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1028" DrawAspect="Content" ObjectID="_1815998166"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;</p><div><div><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1754472441phpzuuZcS.jpeg" alt="" width="371" height="318" loading="lazy"></picture></div></div><p>&nbsp;</p>

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