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Two rotating discs brought into contact: common angular speed or energy lost

This type of question, asked 3 times: 2021 Paper 21, 30 Jan 2024, 8 Apr 2024. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 21

    5
    Two discs have moments of inertia I1I_{1} and I2I_{2} about their respective axes perpendicular to the plane and passing through the centre. They are rotating with angular speeds, ω1\omega_{1} and ω2\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:

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  2. Question 2

    30 Jan 2024

    113
    Two discs of moment of inertia I1=4 kg m2I_{1}= 4\text{ }kg{\text{ }m}^{2} and I2=2 kg m2I_{2}= 2\text{ }kg{\text{ }m}^{2} about their central axes & normal to their planes, rotating with angular speeds 10rad/s10rad/s &4rad/s\& 4rad/s respectively are brought into contact face to face with their axes of rotation coincident. The loss in kinetic energy of the system in the process is ____ J.
  3. Question 3

    30 Jan 2024

    28
    Consider a Disc of mass 5 kg5\text{ }kg, radius 2 m2\text{ }m, rotating with angular velocity of 10rad/s10rad/s about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate together without slipping is  _____\ \_\_\_\_\_J
  4. Question 4

    8 Apr 2024

    96
    A thin circular disc of mass MM and radius RR is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity ω\omega. If another disc of same dimensions but of mass M/2M/2 is placed gently on the first disc co-axially, then the new angular velocity of the system is :

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