A particle is attached to the lower end of a uniform rod which is hinged at its other end as shown in the figure. Another identical particle moving horizontally, collides inelastically and sticks to it. The minimum speed of moving particle so that the rod with particles performs circular motion in a vertical plane will be: [length of rod is $l,$ consider masses of both particles and rod to be same]

Option 1 -

$\sqrt[]{10 g l}$

Option 2 -

$\sqrt[]{20 g l}$

Option 3 -

$\sqrt[]{\frac{70}{3} g l}$

Option 4 -

$\sqrt[]{\frac{175}{3} g l}$

6 Views | Posted a month ago

Asked by Shiksha User

1 Answer

Answered by

alok kumar singh | Contributor-Level 10

a month ago

Correct Option - 3

Detailed Solution:

From conservation of Angular momentum about hinged point

$m v l = (\frac{m l^{2}}{3} + 2 m . l^{2}) ω$

$ω = (\frac{3}{7} \frac{v}{l})$

Now from conservation of Energy.

$\frac{1}{2} \times \frac{7}{3} m l^{2} \times ω^{2} = m g l + 2 m g . 2 l$

$v = \sqrt[]{\frac{70}{3} g l}$

From conservation of Angular momentum about hinged point <math> <mrow> <mi>m</mi> <mi>v</mi> <mi mathvariant="script">l</mi> <mo>=</mo> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mi>m</mi> <msup> <mrow> <mi mathvariant="script">l</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mo>+</mo> <mn>2</mn> <mi>m</mi> <mo>.</mo> <msup> <mrow> <mi mathvariant="script">l</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>ω</mi> </mrow> </math>           <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1819011654"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <mi>ω</mi> <mo>=</mo> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>7</mn> </mrow> </mfrac> <mfrac> <mrow> <mi>v</mi> </mrow> <mrow> <mi mathvariant="script">l</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </math>           <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1819011655"> </o:OLEObject></xml><! [endif]-> Now from conservation of Energy. <math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>×</mo> <mfrac> <mrow> <mn>7</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mi>m</mi> <msup> <mrow> <mi mathvariant="script">l</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>×</mo> <msup> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>m</mi> <mi>g</mi> <mi>l</mi> <mo>+</mo> <mn>2</mn> <mi>m</mi> <mi>g</mi> <mo>.</mo> <mn>2</mn> <mi mathvariant="script">l</mi> </mrow> </math>           <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1819011656"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <mi>v</mi> <mo>=</mo> <mroot> <mrow> <mfrac> <mrow> <mn>7</mn> <mn>0</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mi>g</mi> <mi mathvariant="script">l</mi> </mrow> <mrow></mrow> </mroot> </mrow> </math>           <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1028" DrawAspect="Content" ObjectID="_1819011657"> </o:OLEObject></xml><! [endif]->

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