Question

In order for the ball to be able to make a complete circle around the peg, there must be sufficient speed at the top of its arc such that the centripetal acceleration is large enough to keep the string that. Determine the minimum speed for the ball to achieve a complete circle in terms of L, d, and g. (Hint: use Newton’s 2nd Law and consider what the string tension must be for the ball to just barely get over the top).

Answer 1

Answer:

Explanation:

Let T be the tension in the swing

At top point $mg-T=\frac{mv^2}{r}$

where v=velocity needed to complete circular path

r=distance between point of  rotation to the ball center=L+\frac{d}{2} (d=diameter of ball)

Th-resold velocity is given by $mg-0=\frac{mv^2}{r}$

To get the velocity at bottom conserve energy at Top and bottom

At top $E_T=mg\times 2L+\frac{mv^2}{2}$

Energy at Bottom $E_b=\frac{mv_0^2}{2}$

Comparing two as energy is conserved

$v_0^2=4gr+gr$

$v_0^2=5gr$

$v_0=\sqrt{5gr}$

$v_0=\sqrt{5g\left ( \frac{d}{2}+L\right )}$