HW4_prob3_ hoop on parabola

# HW4_prob3_ hoop on parabola - HW#4 Problem 3 The Rolling...

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HW #4 Problem 3 The Rolling Constraint: A small circular hoop of radius R and mass m rolls without slipping up the inside of the parabola, y = ax 2 . Initially, point 1 is in contact with the origin ( 0 ) of the parabola. After the hoop has rolled up the parabola to the position shown in the left figure, the arc lengths from the current point of contact (point T , indicated by the dashed radial line) to point 1 on the hoop (arc 1T ) and along the parabola from the origin to the current point of contact (arc 0T ) must be equal because slipping is not allowed. From the figure, if we call s the arc length( 0T ), you can see that this condition may be expressed mathematically as s = R ( φ + θ ), where the angle φ (>0) is measured from the solid vertical line to the dashed line and the angle θ (>0) is the angle that measures the hoop’s angular displacement from the inertial (always vertical) reference axis. You could pick another axis as the inertial reference axis, but this axis must always have the same absolute orientation as the hoop rolls on the parabola. It is important to put the rolling constraint in terms of θ because this angle is a good choice for one of the generalized coordinates of this system.

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HW4_prob3_ hoop on parabola - HW#4 Problem 3 The Rolling...

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