a ball rolls around a track that is a circle with a diameter of 170cm in a vertical plane. the ball makes it…

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

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

Answer

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