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 forces at the top

At the top of the loop, the only force providing the centripetal force is gravity. So, \(F_{centripetal}=mg\).

Step2: Use centripetal force formula

The formula for centripetal force is \(F_{centripetal}=\frac{mv^{2}}{r}\). Since \(F_{centripetal}=mg\), we have \(mg = \frac{mv^{2}}{r}\).

Step3: Solve for \(v\)

Cancel out \(m\) from both sides: \(g=\frac{v^{2}}{r}\). Then \(v=\sqrt{gr}\). The radius \(r=\frac{170}{2}=85\) cm. Taking \(g = 980\) cm/s² (approximate value for \(g\) in cm/s²), we get \(v=\sqrt{980\times85}\).

Step4: Calculate the value

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

Answer:

\(288.6\) cm/s