A wheel of radius, b, rolls without slipping at a constant speed, V0. around a c

Question

A wheel of radius, b, rolls without slipping at a constant speed, V0. around a circular track of radius, R. The axle of the wheel is a horizontal rod that turns freely on a pivot at the center of the track. Determine the acceleration of a point at the top of the wheel. (As mentioned in class, it is convenient to place the origin of a coordinate system S at the center of the wheel with the x axis pointing horizontally along the axle toward the pivot at the center of the track and the z axis pointing vertically upward toward the top of the wheel. so that the horizontal y axis is directed opposite to the velocity of the center of the wheel.)

Explanation / Answer

at the top , the acceleration due to rolling

Ac1 = Vo^2/b

and as it is rotating in a horizontal circle

Ac2 = Vo^2/R

Now , net acceleration A = sqrt(Ac1^2 + Ac2^2)

A = Vo^2*(1/b^2 + 1/R^2)

the acceleration of top of wheel is Vo^2*(1/b^2 + 1/R^2)

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