Getting back to the old things #5
The free end of a thread wound on a bobbin of inner radius and outer radius is passed around a nail hammered into the wall.
The thread is then pulled downward at a constant velocity of at an angle of with the vertical.
Find the velocity of the center of mass of the bobbin assuming that it is rolling without slipping at the instant shown in the diagram.
The answer is 32.
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As the thread unwinds, the bobbin rolls on the ground. The bottom most point remains at rest. For no slipping at the bottom most point, we can write, v ′ = 4 R ω
As the string does not slip while unwinding, the velocity of the point P along the string must be equal to the speed of the string. The speed of the point P can also be calculated by considering the rolling motion of the bobbin as a combination of both translatory and rotatory motions.
Therefore, along the string, we can write. R ω − v ′ cos 6 0 ∘ = v
Solving the two equations and substituting the values, we get, v ′ = − 3 2 m/s The negative sign shows that the bobbin will move towards the right.