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Two discs of same moment of inertia rota...

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

A

`(1)/(4) I (omega_1 - omega_2)^(2)`

B

`I (omega_(1) - omega_(2))^(2)`

C

`(1)/(8) I (omega_(1) - omega_(2))^(2)`

D

`(1)/(2) I (omega_(1) + omega_(2))^(2)`

Text Solution

Verified by Experts

The correct Answer is:
A
Initial angular momentum = `I omega_(1) + I omega_(2)`
Let `omega` be angular speed of the combined system .
Final angular momentum = `2 I omega`
Final angular momentum = `2 I omega`
`therefore` According to conservation of angular momentum `I omega_(1) + I omega_(2) = 2 I omega or omega = (omega_(1) + omega_(2))/(2)`
Initial rotational kinetic energy , `E_(1) = (1)/(2) I (omega_(1)^(2) + omega_(2)^(2))`
Final rotational kinetic energy
`E_(f) = (1)/(2) (2 I) omega^(2) = (1)/(2) (2I) ((omega_(1) + omega_(2))/(2))^(2) = (1)/(4) I (omega_(1) + omega_(2))^(2)`
`therefore` Loss of energy `Delta E= E_(i) - E_(f)`
`= (1)/(2) (omega_(1)^(2) + omega_(2)^(2))- (1)/(4) (omega_(1)^(2) + omega_(2) ^(2) + 2omega_(1) omega_(2))`
`= (1)/(4) [ omega_(1)^(2) + omega_(2) ^(2) - 2 omega_(1) omega_(2)] = (1)/(4) (omega_(1) - omega_(2))^(2)`
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Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocity 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 :

Two discs of same moment of inertia rotating 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 the face coinciding the axis of rotation. The expression for loss of enregy during this process is :

Knowledge Check

  • The moment of inertia of a copper disc, rotating about an axis passing through its centre and perpendicular to its plane

    A
    increases if its temperature is increased
    B
    changes if its axis of rotation is changed
    C
    increases if its angular velocity is increased
    D
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  • Two discs of moment of inertia I_(1) and I_(2) about their respectively axes (normal) to the disc and passing through the centre) , and rotating with angular speed omega_(1) and omega_(2) are brought into contact face to face with their axes of rotation coincident . What is the loss in kinetic energy of the system in the process ?

    A
    `(I_(1) I_(2) (omega_(1) - omega_(2))^(2))/(2 (I_(1) + I_(2)))`
    B
    `(I_(1) I_(2) (omega_(1) - omega_2)^(2))/((I_(1) + I_(2)))`
    C
    `(I_(1) I_(2) (omega_(1) + omega_(2))^(2))/((I_(1) - I_(2)))`
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  • Two discs of moments of inertia I_1 and I_2 about their respective axes (normal to the disc and passing through the centre), and rotating with angular speed omega_1 and omega_2 are brought into contact face to face with their axes of rotation coincident. What is the loss in kinetic energy of the system in the process?

    A
    `I_(1)I_(2)(omega_1 - omega_2)^(2)/2(I_1 + I_2)`
    B
    `I_(1)I_(2)(omega_1 - omega_2)^(2)/(I_1 + I_2)`
    C
    `I_(1)I_(2)(omega_1 + omega_2)^(2)/(I_1 - I_2)`
    D
    `I_(1)I_(2)(omega_1 + omega_2)^(2)/2(I_1 - I_2)`

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