Question 671: Centre of Mass and Rotational Motion

Question

A circular disk of moment of inertia $${I_t}$$ is rotating in a horizontal plane, about its symmetry axis, with a constant angular speed $${\omega _i}$$. Another disk of moment of inertia $${I_b}$$ is dropped coaxially onto the rotating disk. Initially the second disk has zero angular speed. Eventually both the disks rotate with a constant angular speed $$\omega $$. The energy lost by the initially  rotating disc to friction is

Accepted Answer

As no external torque is applied to the
system, the angular momentum of the system
remains conserved.
$$ 	herefore $$ L
i
= L
f
According to given problem,
$${I_t}{\omega _i} = \left( {{I_t} + {I_b}} ight){\omega _f}$$
$$ \Rightarrow {\omega _f} = {{{I_t}{\omega _i}} \over {\left( {{I_t} + {I_b}} ight)}}$$    ...(i)
Initial energy, $${E_i} = {1 \over 2}{I_t}\omega _i^2$$   ...(ii)
Final energy,  $${E_f} = {1 \over 2}\left( {{I_t} + {I_b}} ight)\omega _f^2$$   ...(iii)
Substituting the value of $${\omega _f}$$ from equation (i) in equation (iii), we get
Final energy, $${E_f} = {1 \over 2}\left( {{I_t} + {I_b}} ight){\left( {{{{I_t}{\omega _i}} \over {\left( {{I_t} + {I_b}} ight)}}} ight)^2}$$
$$ = {1 \over 2}{{I_t^2\omega _i^2} \over {{I_t} + {I_b}}}$$   ...(iv)
Loss of energy, $$\Delta $$E = E
i
Ã¢ÂÂ E
f
$$ = {1 \over 2}{I_t}\omega _i^2 - {1 \over 2}{{I_t^2\omega _i^2} \over {\left( {{I_t} + {I_b}} ight)}}$$     (Using (ii) and (iv))
$$ = {{\omega _i^2} \over 2}\left( {{I_t} - {{I_t^2} \over {\left( {{I_t} + {I_b}} ight)}}} ight) = {{\omega _i^2} \over 2}\left( {{{I_t^2 - {I_b}{I_t} - I_t^2} \over {\left( {{I_t} + {I_b}} ight)}}} ight)$$
$$ = {1 \over 2}{{{I_b}{I_t}} \over {\left( {{I_t} + {I_b}} ight)}}\omega _i^2$$

Answer

$${1 \over 2}{{I_b^2} \over {\left( {{I_t} + {I_b}} \right)}}\omega _i^2$$

Answer

$${1 \over 2}{{I_t^2} \over {\left( {{I_t} + {I_b}} \right)}}\omega _i^2$$

Answer

$${{{I_b} - {I_t}} \over {\left( {{I_t} + {I_b}} \right)}}\omega _i^2$$

Answer

$${1 \over 2}{{{I_b}{I_t}} \over {\left( {{I_t} + {I_b}} \right)}}\omega _i^2$$

Centre of Mass and Rotational Motion

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