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Question
A disc rotating about its axis with angular speed ωois placed lightly (without any translational push) on a perfectly frictionless table. The radius of the disc is R. What are the linear velocities of the points A, B and C on the disc shown in Figure? Will the disc roll in the direction indicated?
Solution 1
vA = Rωo; vB = Rωo; `v_c = (R/2)omega_o`
The disc will not roll Angular speed of the disc = ωo
Radius of the disc = R
Using the relation for linear velocity, v = ωoR
For point A:
vA = Rωo; in the direction tangential to the right
For point B:
vB = Rωo; in the direction tangential to the left
For point C:
`v_c = (R/2)omega_o` in the direction same as that of vA
The directions of motion of points A, B, and C on the disc are shown in the following figure
Since the disc is placed on a frictionless table, it will not roll. This is because the presence of friction is essential for the rolling of a body.
Solution 2
Since `v = romega`
For Point A, `v_A = Romega_0` in the direction of arrow
For point B, `v_B = Romega_0` in the opposite direction of arrow
For point C, `v_C = R/2omega_0` in the direction of arrow
The disc will not roll in the given direction because friction is necessary for the same
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