A rope of mass m and length L hangs from a ceiling. (1) Show that the wave speed on the rope at a distance y above the lower end is v = √√√gy. (2) Show that the time for a pulse to travel the length of the rope is At = = 2√/L/g.
A rope of mass m and length L hangs from a ceiling. (1) Show that the wave speed on the rope at a distance y above the lower end is v = √√√gy. (2) Show that the time for a pulse to travel the length of the rope is At = = 2√/L/g.
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Question
![**Wave Motion on a Hanging Rope**
Consider a rope of mass \( m \) and length \( L \) suspended from a ceiling.
1. **Wave Speed on the Rope**: Demonstrate that the wave speed at a distance \( y \) above the lower end of the rope is given by the formula:
\[
v = \sqrt{gy}
\]
where \( g \) is the acceleration due to gravity.
2. **Pulse Travel Time**: Show that the time \( \Delta t \) for a pulse to travel the full length of the rope is:
\[
\Delta t = 2\sqrt{\frac{L}{g}}
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Transcribed Image Text:**Wave Motion on a Hanging Rope**
Consider a rope of mass \( m \) and length \( L \) suspended from a ceiling.
1. **Wave Speed on the Rope**: Demonstrate that the wave speed at a distance \( y \) above the lower end of the rope is given by the formula:
\[
v = \sqrt{gy}
\]
where \( g \) is the acceleration due to gravity.
2. **Pulse Travel Time**: Show that the time \( \Delta t \) for a pulse to travel the full length of the rope is:
\[
\Delta t = 2\sqrt{\frac{L}{g}}
\]
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