A strong string of mass 3.00 g and length 2.20 m is tied to supports at each end and is vibrat

step 1 So here they're only asking for the amplitude. So let's find amplitude. Mew. First off, we shoul

A strong string of mass 3.00 g and length 2.20 m is tied...

A strong string of mass $3.00 \mathrm{~g}$ and length $2.20 \mathrm{~m}$ is tied to supports at each end and is vibrating in its fundamental mode. The maximum transverse speed of a point at the middle of the string is $9.00 \mathrm{~m} / \mathrm{s}$ The tension in the string is $330 \mathrm{~N}$. (a) What is the amplitude of the standing wave at its antinode? (b) What is the magnitude of the maximum transverse acceleration of a point at the antinode?

Key Concepts

Amplitude, Angular Frequency, and Maximum Transverse Motion

In oscillatory motion, the amplitude represents the maximum displacement from the equilibrium position. The angular frequency, which is related to the vibrational frequency, plays a crucial role in determining the maximum transverse speed and acceleration at the antinodes of a standing wave. Specifically, the maximum transverse speed is the product of the amplitude and the angular frequency, while the maximum transverse acceleration is the product of the amplitude and the square of the angular frequency.

Wave Speed, Tension, and Linear Mass Density

The wave speed on a string is determined by the tension in the string and its linear mass density, which is the mass per unit length. This relationship is essential for analyzing how the physical properties of the string, such as tension and mass distribution, influence the wavelength and frequency of the standing wave patterns formed on it.

Standing Waves on a String

Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions along a medium and interfere with each other. This results in a vibration pattern characterized by fixed nodes, where the displacement is always zero, and antinodes, where the displacement reaches a maximum. Understanding standing waves is crucial for analyzing vibrational modes in systems like strings that are fixed at both ends.

Fundamental Mode of Vibration

The fundamental mode is the simplest form of vibration in a system and corresponds to the lowest possible frequency, or first harmonic. In a string fixed at both ends, it is the mode where the wavelength is twice the length of the string, and there is one antinode at the center. This concept is key to determining the spatial distribution of motion and the frequencies at which the string vibrates.

Transcript

00:01 So here they're only asking for the amplitude.

00:02 So let's find amplitude.

00:06 Mew.

00:07 First off, we should start off by trying to find the linear density because we have a mass and we have a length.

00:14 So we'll say that mass is going to be equal to 0 .003 kilograms or this is simply three grams just converted into kilograms divided by the full length l of 2 .20 meters.

00:27 And this is going to give us a linear density of 0 .00136 kilograms per meter.

00:35 At this point, the wavelength of the transverse wave of the fundamental, here will be the 2l, so 2 times 2 .20 meters.

00:49 This is going to be 4 .40 meters.

00:53 So that would be the wavelength of the fundamental.

00:55 And then we simply need to find the, we can, rather we can find the speed of propagation now.

01:07 So this will be over f, rather the force tension divided by mu.

01:11 We have a force tension.

01:13 And at this point we just have to substitute.

01:16 And we have 330 newtons divided by the linear density of 0 .00136 kilograms per muons.

01:28 Meter and we find that this is going to be equal to 491 .94 meters per second.

01:37 We also know that the velocity of propagation is going to be equal to the wavelength times the frequency rather.

01:45 And if we solve for the frequency, we can say that this is going to be equal to the velocity divided by the wavelength.

01:52 So we'll have 491 .94 divided by 4 .4 .4 .4 .4 .4 .4 .4 .4 .4 .4...

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