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A uniform disc rotating in a horizontal plane about a vertical axis passing through its centre at 20 r.p.m. has a body of mass of 2 kg kept at its centre. If the mass now slides to the edge, what would the angular speed of the disc be? Mass of the disc = 5 kg. Comment on the change in the KE of the disc if any?
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A uniform disc rotating in a horizontal plane about a vertical axis p...
Problem Statement


  • A uniform disc of mass 5 kg is rotating in a horizontal plane about a vertical axis passing through its center at 20 rpm.

  • A body of mass 2 kg is initially kept at the center of the disc.

  • If the mass slides to the edge of the disc, what would be the angular speed of the disc?



Solution

Let the initial angular speed of the disc be ω1 and the final angular speed be ω2.

Let the radius of the disc be R and the moment of inertia be I.

Given, mass of the disc = 5 kg and mass of the body = 2 kg.


Initial Situation

The initial moment of inertia of the system is given by:

I = ½ mR² + mR² = 1.5mR² = 1.5 x 5 x R² = 7.5R²

The initial angular momentum of the system is given by:

L1 = Iω1 = 7.5R² x (20 x 2π/60) = 5πR²

The initial kinetic energy of the system is given by:

K1 = ½ Iω1² = 3.75πR⁴


Final Situation

The final moment of inertia of the system is given by:

I' = ½ mR² + m(R+R)² = 2.5mR² = 2.5 x 5 x R² = 12.5R²

The final angular momentum of the system is given by:

L2 = I'ω2 = 12.5R² x ω2

Conservation of angular momentum gives:

L1 = L2

5πR² = 12.5R² x ω2

ω2 = 2π x 20 x 5/12.5 = 16π rad/s


Change in Kinetic Energy

The final kinetic energy of the system is given by:

K2 = ½ I'ω2² = 10πR⁴

The change in kinetic energy is given by:

ΔK = K2 - K1 = 6.25πR⁴

The change in kinetic energy is positive, which means that the kinetic energy of the system has increased.


Conclusion

The angular speed of the disc when the mass slides to the edge is 16π rad/s. The kinetic energy of the system increases when the mass slides to the edge.
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A uniform disc rotating in a horizontal plane about a vertical axis passing through its centre at 20 r.p.m. has a body of mass of 2 kg kept at its centre. If the mass now slides to the edge, what would the angular speed of the disc be? Mass of the disc = 5 kg. Comment on the change in the KE of the disc if any?
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