Question 724: Centre of Mass and Rotational Motion

Question

A thin uniform rod, pivoted at O, is rotating in the horizontal plane with constant angular speed $$\omega$$, as shown in the figure. At time t = 0, a small insect starts from O and moves with constant speed v, with respect to the rod towards the other end. It reaches the end of the rod at t = T and stops. The angular speed of the system remains $$\omega$$ throughout. The magnitude of the torque (|$$	au$$|) about O, as a function of time is best represented by which plot ?

Accepted Answer

Let M and l be the mass and length of the rod respectively and m be the mass of the insect. Let the insect be at a distance x from O at any instant of time t.
$$	herefore$$ x = vt ....... (i)
Angular momentum of the system about O,
$$L = \left[ {{{M{l^2}} \over {12}} + m{x^2}} ight]\omega $$
$$ = \left[ {{{M{l^2}} \over {12}} + m{{(vt)}^2}} ight]\omega $$ (Using (i))
$$ = \left[ {{{M{l^2}} \over {12}} + m{v^2}{t^2}} ight]\omega $$
As, $$\left| {\overrightarrow 	au  } ight| = {{dL} \over {dt}} = {d \over {dt}}\left[ {{{M{l^2}} \over {12}} + m{v^2}{t^2}} ight]\omega $$
As $$\omega$$ and v remain constant
$$	herefore$$ $$\left| {\overrightarrow 	au  } ight| = 2m\omega {v^2}t$$
$$\left| {\overrightarrow 	au  } ight| \propto t$$
Hence, the graph and t is a straight line passing through the (0, 0). Option (b) represents correct plot.

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Centre of Mass and Rotational Motion

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