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3 months ago

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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).

1 answer:

kicyunya [3.2K]3 months ago

6 0

Answer:

Explanation:

Let T represent the tension in the swing.

At the peak $mg-T=\frac{mv^2}{r}$

where v denotes the velocity needed to maintain the circular motion.

r equals the distance from the rotation point to the center of the ball, which is L+\frac{d}{2} (with d being the ball's diameter).

The threshold velocity can be expressed as $mg-0=\frac{mv^2}{r}$

To determine the velocity at the bottom, we can use energy conservation principles at both the top and bottom positions.

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

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

By comparing the two states using conservation of energy, we find $v_0^2=4gr+gr$

$v_0^2=5gr$

$v_0=\sqrt{5gr}$

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

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