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Dazed&Confused
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Homework Statement
A child's hoop of mass [itex]M[/itex] and radius [itex]b[/itex] rolls in a straight line with velocity [itex]v[/itex]. Its top is given a light tap with a stick at right angles to the direction of motion. The Impulse of the blow is [itex]I[/itex].
a. Show that this results in a deflection of the line of rolling by angle [itex]\phi = I/Mv [/itex], assuming that the gyroscope approximation holds and neglecting friction with the ground.
b. Show that the gyroscope approximation is valid provided [itex] F << \dfrac{2Mv^2}{b}[/itex], where F is the peak applied force.
The Attempt at a Solution
I don't know for sure the direction of the impulse but assuming it is to the side, that would be the 'angular impulse' would be [itex]Ib[/itex] in the direction perpendicular to the angular momentum [itex]Mbv[/itex] of the wheel, so [itex]\tan\phi = Ib/Mbv = I/Mv ≈ \phi [/itex]. The last part using the small angle approximation.
For b. I would assume that since [itex]\phi << 1[/itex], therefore [itex] Fdt << Mv [/itex].
I don't know if any of my attempt is correct.
Edit: if instead I approximate the impulse by a triangle height [itex]F[/itex] and base [itex] dt [/itex], then the impulse would be [itex]\dfrac{Fdt}{2}[/itex].
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