Two discs have moments of intertia $I_{1}$ and $I_{2}$ about their respective axes perpendicular to the plane and passing through the centre. They are rotating with angular speeds, $\omega_{1}$ and $\omega_{2}$ respectively and are brought into contact face to face with their axes of rotation coaxial. The loss in kinetic energy of the system in the process is given by:

(C) $\frac{I_{1} I_{2}}{2\left(I_{1}+I_{2}\right)}\left(\omega_{1}-\omega_{2}\right)^{2}$

(A) $\frac{\left(I_{1}-I_{2}\right)^{2} \omega_{1} \omega_{2}}{2\left(I_{1}+I_{2}\right)}$

(B) $\frac{I_{1} I_{2}}{\left(I_{1}+I_{2}\right)}\left(\omega_{1}-\omega_{2}\right)^{2}$

(D) $\frac{\left(\omega_{1}-\omega_{2}\right)^{2}}{2\left(I_{1}+I_{2}\right)}$

JEE Main · 2021 · 27 Aug 2021 · Shift 2 · Rotational Mechanics

Two discs have moments of intertia

I_{1}

and

I_{2}

about their respective axes perpendicular to the plane and passing through the centre. They are rotating with angular speeds,

ω_{1}

and

ω_{2}

respectively and are brought into contact face to face with their axes of rotation coaxial. The loss in kinetic energy of the system in the process is given by:

(A) $\frac{( I _{1} - I _{2} ) ^{2} ω _{1} ω _{2}}{2 ( I _{1} + I _{2} )}$
(B) $\frac{I _{1} I _{2}}{( I _{1} + I _{2} )} (ω_{1} - ω_{2})^{2}$
(C) $\frac{I _{1} I _{2}}{2 ( I _{1} + I _{2} )} (ω_{1} - ω_{2})^{2}$ ✓ Correct
(D) $\frac{( ω _{1} - ω _{2} ) ^{2}}{2 ( I _{1} + I _{2} )}$

Answer: (C)

Solution

Angular momentum is conserved as the discs couple:

I_{1} ω_{1} + I_{2} ω_{2} = (I_{1} + I_{2}) ω_{f}

, so

ω_{f} = \frac{I _{1} ω _{1} + I _{2} ω _{2}}{I _{1} + I _{2}}

. The kinetic-energy loss is

Δ K = (\frac{1}{2} I_{1} ω_{1}^{2} + \frac{1}{2} I_{2} ω_{2}^{2}) - \frac{1}{2} (I_{1} + I_{2}) ω_{f}^{2} .

Substituting

ω_{f}

and simplifying the algebra collapses the cross terms to

Δ K = \frac{I _{1} I _{2}}{2 ( I _{1} + I _{2} )} (ω_{1} - ω_{2})^{2} .

The loss in kinetic energy is

\frac{I _{1} I _{2}}{2 ( I _{1} + I _{2} )} (ω_{1} - ω_{2})^{2}

JEE Main · 2021 · 27 Aug 2021 · Shift 2 · Single correct option

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JEE Main · 2021 · 27 Aug 2021 · Shift 2 · Rotational Mechanics

Solution

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