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Rotational Motion question

2016 · Q147
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Rotational Motion question

2016 · Q147

NEETPhysicsRotational MotionMCQ+4 / −1
A disc and a sphere of same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first?
  1. A
    Both reach at the same time
  2. B
    Depends on their masses
  3. C
    Disc
  4. D
    Sphere
View written solutionFree

Correct answer: D

Time taken by the body to reach the bottom when it rolls down on an inclined plane without slipping is given by

t=2l(1+k2R2)gsin⁡θt = \sqrt {{{2l\left( {1 + {{{k^2}} \over {{R^2}}}} \right)} \over {g\sin \theta }}} t=gsinθ2l(1+R2k2​)​​

Since g is constant and lll, R and sin θ\theta θ are same for both

∴\therefore∴ tdts=1+kd2R21+ks2R2=1+R22R21+2R25R2{{{t_d}} \over {{t_s}}} = {{\sqrt {1 + {{k_d^2} \over {{R^2}}}} } \over {\sqrt {1 + {{k_s^2} \over {{R^2}}}} }} = \sqrt {{{1 + {{{R^2}} \over {2{R^2}}}} \over {1 + {{2{R^2}} \over {5{R^2}}}}}} ts​td​​=1+R2ks2​​​1+R2kd2​​​​=1+5R22R2​1+2R2R2​​​

(∵kd=R2,ks=25R) \left(\because {{k_d} = {R \over {\sqrt 2 }},{k_s} = \sqrt {{2 \over 5}} R} \right)(∵kd​=2​R​,ks​=52​​R)

⇒32×57=1514⇒td>ts \Rightarrow \sqrt {{3 \over 2} \times {5 \over 7}} = \sqrt {{{15} \over {14}}} \Rightarrow {t_d} \gt {t_s}⇒23​×75​​=1415​​⇒td​>ts​

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