A bicycle wheel is mounted on a fixed, frictionless axle, as shown ( Figure 1 )

Question

A bicycle wheel is mounted on a fixed, frictionless axle, as shown (Figure 1) . A massless string is wound around the wheel's rim, and a constant horizontal force F?  of magnitude F starts pulling the string from the top of the wheel starting at time t=0when the wheel is not rotating. Suppose that at some later time t the string has been pulled through a distance d. The wheel has moment of inertia Iw=kmr2, where k is a dimensionless number less than 1, m is the wheel's mass, and r is its radius. Assume that the string does not slip on the wheel.

The force F?  pulling the string is constant; therefore the magnitude of the angular acceleration ? of the wheel is constant for this configuration.

Find the magnitude of the angular velocity ? of the wheel when the string has been pulled a distance d.

Note that there are two ways to find an expression for ?; these expressions look very different but are equivalent.

Express the angular velocity ? of the wheel in terms of the displacement d, the magnitude Fof the applied force, and the moment of inertia of the wheel Iw, if you've found such a solution. Otherwise, following the hints for this part should lead you to express the angular velocity ? of the wheel in terms of the displacement d, the wheel's radius r, and ?.

Explanation / Answer

You already answered part (a) correctly

(b) since alpha is constant
w = alpha * t
d = r * (alpha * t^2) / 2

hence
t = sqrt(2d / (alpha * r))
w = sqrt( 2d * alpha / r) = sqrt( 2d * F / I)

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