Question

A 5.1-m-long rope of mass 2.0 kg hangs from a ceiling.

Part A:What is the tension in the rope at the bottom end?
Part B:What is the wave speed in the rope at the bottom end?
Part C:What is the tension in the rope at the top end, where it is attached to the ceiling?
Part D:What is the wave speed in the rope at the top end?
Part E:It can be shown that the average wave speed in the rope is 12gL−−−√, where L is the length of the rope. Use the average wave speed to calculate the time required for a pulse produced at the bottom of the rope to travel to the top.

Answer 1

Final Answer:

Part A: The tension in the rope at the bottom end is equal to the weight of the rope, given by T = mg, where m is the mass and g is the acceleration due to gravity.

Part B: The wave speed in the rope at the bottom end can be determined using the formula v = √(T/μ), where T is the tension and μ is the mass per unit length of the rope.

Part C: The tension in the rope at the top end, where it is attached to the ceiling, is equal to the sum of the weight of the rope and the tension at the bottom end.

Part D: The wave speed in the rope at the top end can be calculated using the same formula as in Part B, using the tension at the top end.

Part E: Using the given formula for the average wave speed, the time required for a pulse to travel from the bottom to the top of the rope can be determined.

Step-by-step explanation:

The tension at the bottom end (Part A) is the weight of the rope, calculated as T = mg, where m is the mass and g is the acceleration due to gravity. For Part B, the wave speed at the bottom end is given by v = √(T/μ), where T is the tension and μ is the mass per unit length of the rope.

Moving to Part C, the tension at the top end involves both the weight of the rope and the tension at the bottom end. Part D uses the same wave speed formula as in Part B, but with the tension at the top end.

Lastly, Part E utilizes the provided formula for average wave speed to find the time it takes for a pulse to travel from the bottom to the top of the rope.

Answer 2

The tension in the rope is 19.6 N. At both the bottom and top ends, the wave speed is 48.22 m/s. Using the average wave speed of 44.35 m/s, it takes a pulse 0.115 s to travel from the bottom of the rope to the top.

Step-by-step explanation:

The student's question pertains to the physics of waves in a rope, analyzing tension and wave speed at different points in the rope. The tension and wave speed depend on the linear mass density (μ), the force due to gravity, and the mass of the object.

Part A: Tension at the Bottom End

The tension at the bottom end is equal to the weight of the rope since there are no other forces acting on that point. Given that the rope has a mass of 2.0 kg and using the acceleration due to gravity (g = 9.8 m/s²), tension can be calculated as:

T = m × g = 2.0 kg × 9.8 m/s² = 19.6 N.

Part B: Wave Speed at the Bottom End

Wave speed (v) can be calculated using the formula v = √(T/μ), where T is the tension and μ is the linear mass density. At the bottom end, the wave speed is:

v = √(19.6 N / 0.0085 kg/m) = 48.22 m/s.

Part C: Tension at the Top End

The tension at the top end is the same as at the bottom end in a stationary rope without any additional forces, which is 19.6 N.

Part D: Wave Speed at the Top End

Similarly, the wave speed at the top will also be the same as at the bottom (μ is uniform along the rope), which is 48.22 m/s.

Part E: Time for Pulse to Travel from Bottom to Top

The average wave speed (ν) according to the formula provided is ν = 12gL√, replacing L with the length of the rope. Calculating this gives:

ν = 12 × 9.8 m/s² × 5.1 m√ = 44.35 m/s.

The time (t) it takes for a pulse to travel to the top can be calculated by dividing the length of the rope (L) by the average wave speed (ν):

t = L/ν = 5.1 m / 44.35 m/s = 0.115 s