Two discs of moment of inertia I1 and I2 and angular speeds ω 1 and ω 2 are rotating along collinear axes passing through their centre of mass and perpendicular to their plane. If the two are made to rotate combindly along the same axis the rotational KE of system will be

Question

Two discs of moment of inertia I1 and I2 and angular speeds ω 1 and ω 2 are rotating along collinear axes passing through their centre of mass and perpendicular to their plane. If the two are made to rotate combindly along the same axis the rotational KE of system will be (A)  (I1ω 1+I2ω 2/2(I1+I2))  (B)  ((I1+I2) (ω 1+ω 2)2/2)  (C)  ((I1ω I2+ (I2ω 2)2/2(I1+I2))  (D) None of these

Tardigrade Admin · Accepted Answer

Conservation of angular momentum I1ω 1+I2ω 2=(I1+I2)ω Angular velocity of system ω =(I1ω 1+I2ω 2/I1+I2) ∴ Rotational kinetic energy =(1/2)(I1+I2)ω 2 =(1/2)(I1+I2)( (I1ω 1+I2ω 2/I1+I2) )2 =((I1ω 1+I2ω 2)2/2(I1+I2))

Q. Two discs of moment of inertia $I_{1}$ and $I_{2}$ and angular speeds $ω_{1}$ and $ω_{2}$ are rotating along collinear axes passing through their centre of mass and perpendicular to their plane. If the two are made to rotate combindly along the same axis the rotational KE of system will be

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

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

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

None of these

Q. Two discs of moment of inertia I1​ and I2​ and angular speeds ω1​ and ω2​ are rotating along collinear axes passing through their centre of mass and perpendicular to their plane. If the two are made to rotate combindly along the same axis the rotational KE of system will be

2(I1​+I2​)I1​ω1​+I2​ω2​​

2(I1​+I2​)(ω1​+ω2​)2​

2(I1​+I2​)(I1​ωI2​+(I2​ω2​)2​

None of these

Conservation of angular momentum I1​ω1​+I2​ω2​=(I1​+I2​)ω Angular velocity of system ω=I1​+I2​I1​ω1​+I2​ω2​​ ∴ Rotational kinetic energy =21​(I1​+I2​)ω2 =21​(I1​+I2​)(I1​+I2​I1​ω1​+I2​ω2​​)2 =2(I1​+I2​)(I1​ω1​+I2​ω2​)2​

Q. Two discs of moment of inertia $I_{1}$ and $I_{2}$ and angular speeds $ω_{1}$ and $ω_{2}$ are rotating along collinear axes passing through their centre of mass and perpendicular to their plane. If the two are made to rotate combindly along the same axis the rotational KE of system will be

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

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

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