Question

a ball rolls around a track that is a circle with a diameter of 86cm 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: Find radius of the track

The diameter is 86 cm, so radius $r = \frac{86}{2} = 43$ cm. Convert to meters: $r = 0.43$ m.

Step2: Set up force equilibrium

At the top, gravity provides centripetal force: $mg = m\frac{v^2}{r}$. Cancel $m$: $g = \frac{v^2}{r}$.

Step3: Solve for speed $v$

Rearrange formula: $v = \sqrt{gr}$. Use $g = 9.8\ \text{m/s}^2$.
$v = \sqrt{9.8 \times 0.43}$
$v = \sqrt{4.214} \approx 2.053\ \text{m/s}$

Step4: Convert to cm/s

Multiply by 100: $2.053 \times 100 = 205.3$ cm/s.

Answer:

205.3 cm/s