Question 638: Centre of Mass and Rotational Motion

Question

Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities $${\omega _1}$$ and $${\omega _2}$$.  They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is

Accepted Answer

According to the problem,
I$${\omega _1}$$ + I$${\omega _2}$$ = 2I$${\omega _0}$$
$${\omega _0}$$ = $${{\left( {{\omega _1} - {\omega _2}} \right)} \over 2}$$
$${K_i} = {1 \over 2}I\left( {{\omega _1}^2 + {\omega _2}^2} \right)$$
$${K_f} = {1 \over 4}{\left( {{\omega _1} + {\omega _2}} \right)^2}$$
Loss $$\Delta K = I\left[ {{{{\omega _1}^2} \over 2} + {{{\omega _2}^2} \over 2} - {{{\omega _1}^2} \over 4} - {{{\omega _2}^2} \over 4} - {{2{\omega _1}{\omega _2}} \over 4}} \right]$$
$$ = I\left[ {{{{\omega _1}^2} \over 4} + {{{\omega _2}^2} \over 4} - {{2{\omega _1}{\omega _2}} \over 4}} \right]$$
$$ = {I \over 4}{\left[ {{\omega _1} - {\omega _2}} \right]^2}$$

Answer

$${1 \over 4}I{\left( {{\omega _1} - {\omega _2}} \right)^2}$$

Answer

$$I{\left( {{\omega _1} - {\omega _2}} \right)^2}$$

Answer

$${1 \over 8}I{\left( {{\omega _1} - {\omega _2}} \right)^2}$$

Answer

$${1 \over 2}I{\left( {{\omega _1} + {\omega _2}} \right)^2}$$

Centre of Mass and Rotational Motion

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