KVPY Sample Paper KVPY Stream-SX Model Paper-29

  • question_answer
    A uniform rod of mass m, length S is placed over a smooth horizontal surface along y-axis and is at rest as shown in figure- An impulsive force F is applied for a small time At along positive x-direction at end A of the rod. The x-coordinate of end A of the rod when the rod becomes parallel to x-axis for the first time is (initially the coordinate of centre of mass of the rod is (0,0)):

    A) \[\frac{\pi \ell }{12}\]

    B) \[\frac{\ell }{2}\left( 1+\frac{\pi }{12} \right)\]

    C) \[\frac{\ell }{2}\left( 1-\frac{\pi }{6} \right)\]

    D) \[\frac{\ell }{2}\left( 1+\frac{\pi }{6} \right)\]

    Correct Answer: D

    Solution :

    As torque = change om angular momentum
    \[\therefore F,\,\,\Delta t=mv\]                              (Linear)... (1)
    and \[\left( F.\frac{\ell }{2} \right)\Delta t=\frac{m{{\ell }^{2}}}{12}.\omega \]                  (angular)...(2)
    Dividing:            (1) and (2)
    \[2=\frac{12v}{\omega \ell }\Rightarrow \,\,\,\,\,\,\,\,\,\omega =\frac{6v}{\ell }\]
    Using : S = ut:
    Displacement of COM is: \[\frac{\pi }{2}=\omega t=\left( \frac{6v}{\ell } \right)t\]
    and       \[x=vt\]
    Dividing: \[\frac{2x}{\pi }=\frac{\ell }{6}\]           \[\Rightarrow \,\,\,\,x=\frac{\pi \ell }{12}\]\[\Rightarrow \]            Coordination of A will be \[\left[ \frac{\pi \ell }{12}+\frac{\ell }{2},0 \right]\]
    Hence [D].

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