• VP3. A pendulum, comprising a light string of length L and a small sphere, swings in the vertical plane. The string hits a peg located a distance d below the point of suspension (see figure above). (a) Show that if the sphere is released from a height below that of the peg, it will return to thic hoight aft the a strike +he

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Ch7 VP3
**VP3.** A pendulum, comprising a light string of length \( L \) and a small sphere, swings in the vertical plane. The string hits a peg located a distance \( d \) below the point of suspension (see figure above). 

(a) Show that if the sphere is released from a height below that of the peg, it will return to this height after the string strikes the peg. 

(b) Show that if the pendulum is released from rest at the horizontal position (\( \theta = 90^\circ \)) and is to swing in a complete circle centered on the peg, the minimum value of \( d \) must be \( 3L/5 \).

**Diagram Explanation:**

- The diagram presents a pendulum setup with a peg inserted below the suspension point.
- The pendulum string has a length \( L \).
- The angle \( \theta \) is defined between the vertical line and the string.
- The distance \( d \) is indicated from the point of suspension down to the peg.
- The pendulum's path, after interacting with the peg, is illustrated with a dashed arc.
Transcribed Image Text:**VP3.** A pendulum, comprising a light string of length \( L \) and a small sphere, swings in the vertical plane. The string hits a peg located a distance \( d \) below the point of suspension (see figure above). (a) Show that if the sphere is released from a height below that of the peg, it will return to this height after the string strikes the peg. (b) Show that if the pendulum is released from rest at the horizontal position (\( \theta = 90^\circ \)) and is to swing in a complete circle centered on the peg, the minimum value of \( d \) must be \( 3L/5 \). **Diagram Explanation:** - The diagram presents a pendulum setup with a peg inserted below the suspension point. - The pendulum string has a length \( L \). - The angle \( \theta \) is defined between the vertical line and the string. - The distance \( d \) is indicated from the point of suspension down to the peg. - The pendulum's path, after interacting with the peg, is illustrated with a dashed arc.
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