Two discs of moments of inertia I1 and I2 about their respective axes, rotating with angular frequencies, ω1 and ω2 respectively, are brought into contact face to face with their axes of rotation coincident. The angular frequency of the composite disc will be A

Question

Two discs of moments of inertia I1 and I2 about their respective axes, rotating with angular frequencies, ω1 and ω2 respectively, are brought into contact face to face with their axes of rotation coincident. The angular frequency of the composite disc will be A (A) (I1 ω1+I2 ω2/I1+I2) (B) (I2 ω1+I1 ω2/I1+I2) (C) (I1 ω1+I2 ω2/I1-I2) (D) (I2 ω1+I1 ω2/I1-I2)

Tardigrade Admin · Accepted Answer

Total initial angular momentum of the two discs is

L_{i}

= I_{1} ω_{1} + I_{2} ω_{2}

When two discs are brought into contact face to face (one on top of the other) and their axes of rotation coincide, the moment of inertia I of the system is equal to the sum of their individual moments of inertia.
i.e.

I = I_{1} + I_{2}

Let

ω

be the final angular speed of the system.
The final angular momentum of the system is

L_{f} = I ω = (I_{1} + I_{2}) ω

As no external torque acts on the system, therefore according to law of conservation of angular momentum, we get

L_{i} = L_{f}

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

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

Q. Two discs of moments of inertia $I_{1}$ and $I_{2}$ about their respective axes, rotating with angular frequencies, $ω_{1}$ and $ω_{2}$ respectively, are brought into contact face to face with their axes of rotation coincident. The angular frequency of the composite disc will be $A$

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

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

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

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

Q. Two discs of moments of inertia I1​ and I2​ about their respective axes, rotating with angular frequencies, ω1​ and ω2​ respectively, are brought into contact face to face with their axes of rotation coincident. The angular frequency of the composite disc will be A

I1​+I2​I1​ω1​+I2​ω2​​

I1​+I2​I2​ω1​+I1​ω2​​

I1​−I2​I1​ω1​+I2​ω2​​

I1​−I2​I2​ω1​+I1​ω2​​

Q. Two discs of moments of inertia $I_{1}$ and $I_{2}$ about their respective axes, rotating with angular frequencies, $ω_{1}$ and $ω_{2}$ respectively, are brought into contact face to face with their axes of rotation coincident. The angular frequency of the composite disc will be $A$

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

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

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

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