Question

a ball rolls around a track that is a circle with a diameter of 170cm in a vertical plane. the ball makes it all the way around, but only just barely - at the moment when it is at the top, the track has no effect at all - the ball is effectively a projectile at that moment. determine the speed of the ball in centimeters per second when it is at the top of the loop. provide at least one decimal place

Explanation:

Step1: Analyze the forces at the top

At the top of the loop, the only force providing the centripetal force is gravity. Using the centripetal force formula \(F_c=\frac{mv^{2}}{r}\), and since \(F_c = mg\) (at the critical point where the track has no effect), we have \(mg=\frac{mv^{2}}{r}\).

Step2: Solve for velocity

Cancel out the mass \(m\) from both sides of the equation \(mg=\frac{mv^{2}}{r}\). We get \(v = \sqrt{gr}\). The diameter \(d = 170cm\), so the radius \(r=\frac{d}{2}=85cm\). Taking \(g = 980cm/s^{2}\), then \(v=\sqrt{980\times85}\).

$$v=\sqrt{980\times85}=\sqrt{83300}\approx288.6cm/s$$

Answer:

\(288.6cm/s\)