A hoop with a mass \(m\) and unknown radius is rolling without slipping on a flat surface with an angular speed \(\omega\). The hoop encounters a hill and continues to roll without slipping until it reaches a maximum height \(h\).(a) Derive an expression for the radius of the hoop in terms of given quantities and fundamental constants. , (b) What direction does friction point while the hoop is rolling up the ramp? Explain your reasoning. , (c) After reaching the maximum height, the hoop begins to roll back down. What direction does friction point while the hoop is rolling down the ramp? Explain your reasoning. :

\( R = \sqrt{\frac{gh}{\omega^2}} \) Up the ramp. Up the ramp.

A hoop with a mass \(m\) and unknown radius is rolling without slipping on a flat surface with an angular speed \(\omega\). The hoop encounters a hill and continues to roll without slipping until it reaches a maximum height \(h\). • Nerd Notes

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AP Physics 1

6.5 Rolling

6.1 Rotational Kinetic Energy

5.6 Newton’s Second Law in Rotational Form

5.4 Rotational Inertia

3.4 Conservation of Energy

2.7 Kinetic and Static Friction

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AdvancedFRQMathematicalConceptual15.7k

Figure 1. A hoop rolls left on a flat surface, then rolls up an incline to height \(h\).

A hoop with a mass \(m\) and unknown radius is rolling without slipping on a flat surface with an angular speed \(\omega\). The hoop encounters a hill and continues to roll without slipping until it reaches a maximum height \(h\).

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