A 75 g bungee cord has an equilibrium length of 1.20 m. The...

A 75 g bungee cord has an equilibrium length of $1.20 \mathrm{m}$. The cord is stretched to a length of $1.80 \mathrm{m},$ then vibrated at $20 \mathrm{Hz}$ This produces a standing wave with two antinodes. What is the spring constant of the bungee cord?

Key Concepts

Hooke's Law

Hooke's Law states that the force required to extend or compress a spring by some distance is proportional to that distance, expressed as F = kx. This principle allows us to relate the tension in a spring to its extension and determine the spring constant, which measures the spring's stiffness.

Standing Waves

Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere. In a system with fixed ends, the boundary conditions cause nodes at the ends and determine the possible modes of vibration. The number of antinodes (points of maximum displacement) indicates the harmonic being excited, which in turn is used to relate the vibrating length to the wave’s frequency.

Wave Speed on a Tensioned Medium

The speed of a wave traveling on a string under tension is given by v = ?(T/?), where T is the tension and ? is the linear mass density of the string. This relationship is key in linking the dynamic behavior of the string (its frequency and wavelength in standing wave patterns) to the static tension that develops due to the extension of the spring.

Linear Density

Linear density (?) is defined as the mass per unit length of a string or cord. It is calculated by dividing the total mass by its current length. The value of ? is fundamental in determining the wave speed along the string, and is thus essential in analyzing the behavior of vibrating systems.

Transcript

00:01 This question is about a bungee cord, and i know that a bungee cord acts like a spring.

00:08 Force equals kx, where x is the stretch.

00:22 Now, we are told that mu, basically, is 0 .075 kilograms.

00:37 Per 1 .20 meters.

00:45 So that is our linear density.

00:51 I'm also going to use, i'm going to write down that l original is 1 .20 meters, but it's stretched to 1 .8 meters.

01:09 So that means that the stretch, would be 1 .8 minus 1 .2, which would be 0 .60 meters.

01:25 And so the final length would be l plus x, which is the 1 .8.

01:34 So this length is vibrated at a frequency of 20 hertz, and it produces, a standing wave with two antinodes.

01:53 So i'm gonna draw two antinodes here with the vibration in here.

02:11 Okay, that's showing me that the mode is, so the fundamental frequency is just going to be half of the 20.

02:40 And it goes two right here.

02:43 Going to be 10 hertz.

02:47 And i could actually write f sub 2 here because we know it's the second harmonic.

02:55 All right.

02:59 The fundamental frequency is v over to l, where l in this case is actually l plus x.

03:13 It's the length when it's vibrating.

03:18 V is going to be the square root of t over mu.

03:28 We know what mu is, but t would be opposing the stretching force...

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