Bungee chord (standing waves and antinodes.)

In summary, to find the spring constant of a 75 g bungee cord with an equilibrium length of 1.20 m that is stretched to 1.80 m and vibrated at 20 Hz with two antinodes, you can use the equation f=nv/2L to determine the velocity, then use v^2 = F/(linear mass density) to find the tension. From there, you can use F = kx and solve for k to find the spring constant, which in this case is approximately 40N/m.
  • #1
j88k
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Homework Statement



A 75 g bungee cord has an equilibrium length of 1.20 m. The cord is stretched to a length of 1.80 m, then vibrated at 20 Hz. This produces a standing wave with two antinodes.


What is the spring constant of the bungee cord?



Homework Equations



f=nv/2L


The Attempt at a Solution



Two anti-nodes mean 3 harmonics
f = 20Hz
n = number of harmonics
L = length
v = velocity
20 = nv/(2L)=3v/(2L)
v = 20/3*2*1.8 =24m/s

F = tension
v^2 = F/(linear mass density)
F = 24^2 * 0.075/1.8
F = 24N
F = kx
k =F/x
k = 24/(1.8-1.2) = 40N/m
i
It doesn't seem to be right though!
 
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  • #2
I didn't check your solution numerically, but I agree 100% with your approach.
 

1. What are standing waves and antinodes?

Standing waves are created when a wave reflects off a fixed boundary and interferes with its original wave. They are characterized by nodes, which are points that do not move, and antinodes, which are points that experience the maximum displacement. In a bungee chord, the fixed boundary is the attachment point to a bridge or other structure.

2. How are standing waves and antinodes related to bungee jumping?

When a person jumps off a bridge with a bungee chord attached, they experience the forces of both gravity and the tension from the bungee chord. As the person falls, the bungee chord stretches and creates standing waves with antinodes at certain points along the chord. These antinodes help absorb and distribute the energy of the falling person, making the experience safer.

3. What factors affect the formation of standing waves and antinodes in a bungee chord?

The length of the bungee chord, the tension in the chord, and the frequency of the wave all affect the formation of standing waves and antinodes. A longer chord will result in more antinodes, and a higher tension or frequency will result in smaller distances between the antinodes.

4. Are there any risks associated with standing waves and antinodes in bungee jumping?

While standing waves and antinodes help absorb and distribute the energy of a falling person, they can also amplify the forces acting on the body. This can potentially lead to injuries if the bungee chord is not designed or used properly. It is important for bungee jumpers to always follow safety guidelines and use equipment that is in good condition.

5. Can standing waves and antinodes be observed in other scenarios besides bungee jumping?

Yes, standing waves and antinodes can be observed in many different scenarios, such as musical instruments, microwave ovens, and even ocean waves. They are a fundamental concept in wave mechanics and can be found in various systems where waves are present.

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