A stationary bicycle wheel of radius 0.6 m is mounted in the vertical plane (see

Question

A stationary bicycle wheel of radius 0.6 m is mounted in the vertical plane (see figure below). The axle is held up by supports that are not shown, and the wheel is free to rotate on the nearly frictionless axle. The wheel has mass 4.2 kg, all concentrated in the rim (the spokes have negligible mass). A lump of clay with mass 0.5 kg falls and sticks to the outer edge of the wheel at the location shown. Just before the impact the clay has speed 6 m/s, and the wheel is rotating clockwise with angular speed 0.30 rad/s. (Assume +x is to the right, +y is upward, and +z is out of the page. Assume the line connecting the center to the point of impact is at an angle of 45 degree from the horizontal.) Just before the impact, what is the angular momentum (magnitude and direction) of the combined system of wheel plus clay about the center C? magnitude 1.044 kg · m^2/s direction +z Just after the impact, what is the angular momentum (magnitude and direction) of the combined system of wheel plus clay about the center C? magnitude kg · m^2/s direction +z Just after the impact, what is the angular velocity (magnitude and direction) of the wheel? Magnitude 0.936 rad/s direction +z

Explanation / Answer

a) L = m v r = 0.5 * 6 * 0.6 * sin 45

= 1.27 kg.m2/s

b)  As there is no torque on the system

L = m v r = 0.5 * 6 * 0.6 * sin 45

= 1.27 kg.m2/s

c) w = m v cos theta / (m + M)R

= (0.5 * 6 * cos 45) / (0.5 + 4.2) 0.6

= 0.752 rad/s

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