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, ω1 and ω2 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:

Question

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, ω1 and ω2 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) ( I 1 I 2/( I 1+ I 2))(ω1-ω2)2 (B) (( I 1- I 2)2 ω1 ω2/2( I 1+ I 2)) (C) ( I 1 I 2/2( I 1+ I 2))(ω1-ω2)2 (D) ((ω1-ω2)2/2( I 1+ I 2))

Tardigrade Admin · Accepted Answer

From conservation of angular momentum we get I 1 ω1+ I 2 ω2=( I 1+ I 2) ω ω=( I 1 ω1+ I 2 ω2/ I 1+ I 2) k i =(1/2) I 1 ω12+(1/2) I 2 ω22 k f =(1/2)( I 1+ I 2) ω2 k i - k f =(1/2)[ I 1 ω12+ I 2 ω22-(( I 1 ω1+ I 2 ω2)2/ I 1+ I 2)] Solving above we get k i - k f =(1/2)(( I 1 I 2/ I 1+ I 2))(ω1-ω2)2

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

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

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

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

(I1​+I2​)I1​I2​​(ω1​−ω2​)2

2(I1​+I2​)(I1​−I2​)2ω1​ω2​​

2(I1​+I2​)I1​I2​​(ω1​−ω2​)2

2(I1​+I2​)(ω1​−ω2​)2​

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

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

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

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