A metal bar has a length of 1.5 m and a density of 6400 kg/m 3 . Sound waves take 3.9 × 10 -4 s to travel along the length of the bar. What is Young's modulus for this metal?

Welcome back, everyone. Hopefully, you try this practice problem on your own. So in this problem, we're told that a metal bar has a length of 1 and a half meters and a density of 64100. So if I imagine this sort of metal pipe like this, then I've got a length of the bar, which is 1.5. I'm gonna call this l. And I'm told what the density of this material is, which is 64100. I'm also told that sound waves take a certain amount of time to travel along the length of the bar. So if I were to hit this, the end of this metal pipe here, then it would take approximately 3.9 times 10 to the minus 4 seconds to travel to the other side. So that's gonna be a delta t. So what I'm looking for in this problem is I'm asked to find what the Young's modulus of the of this material is. So remember Young's modulus is this sort of Greek letter that kind of looks like a y here. Right? So we're looking really for what is that Young's modulus. Now that really pops up only in one equation, which is gonna be the velocity equation for the longitudinal wave in a solid. Remember that we have liquids and solids with longitudinal waves, and and the difference is we're gonna use the bulk modulus for liquids and the Young's modulus for solid. So this is really the equation we're gonna look on look at. So let's start off with that equation here. We have v equals the square roots of that Young's modulus divided by the density of the material. If I'm looking for Young's modulus, then I just need those other two variables. So do I have what the density is? Yes. I clearly do because that's just 64100, so that's fine. What I don't know here is I don't know what the velocity of the wave is, at least I'm not told explicitly what that is. So in order to solve this problem, I'm gonna have to go find velocity, elsewhere. So how do I do that? Well, let's start with the basic equation for velocity. Remember that velocity is always just calculated as delta x over delta t, displacement over time. So what's the velocity of a sound wave here? Well, really, what happens is if it takes 3.9 times 10 to the minus 4 seconds, travel along the length of the bar, that's delta t. And what's the length of the bar? It's really just the l that I'm given. It's the 1.5 meters. So really this is just l divided by delta t, so I can just go calculate that first. So this is gonna be 1.5 divided by 3.9 times 10 to the minus 4. And if you go plug this in, what you're gonna get is 3,846 meters per second. This is an incredibly fast velocity here, but this is actually very accurate for something like a metal. Metal, sounds travel very fast through metals and liquids, about 10 times faster than they do through air. So now that I have this velocity, I really just plug this back into this equation so I can solve for Young's modulus here. So what I have is v squared equals, because I have to square both sides to get rid of the square roots, is Young's modulus over the density. So, finally, all I have to do to isolate this Young's modulus is I just move the density to the other side. So really what I happy what I have here is that Young's modulus is equal to v squared times rho. So in other words, this is gonna be 3846 times the density of the material 64100, but I have to square the velocity. And what you end up beginning when you multiply this out is you're gonna get something like about 9 point 47 times 10 to the 10th. And the units for Young's modules are gonna be in pascals. So that is the answer to our problem. And if you look at your answer choices, this will be answer choice b. Alright. So pretty straightforward question. Let me know if you have any questions, and I'll see you in the next one.

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18. Waves & Sound

Velocity of Longitudinal Waves

Physics

18. Waves & Sound

Velocity of Longitudinal Waves

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Multiple Choice

A metal bar has a length of 1.5 m and a density of 6400 kg/m³. Sound waves take 3.9 × 10^-4 s to travel along the length of the bar. What is Young's modulus for this metal?

2,311 Pa

9.47 × 10¹⁰ Pa

2.46 × 10⁷ Pa

3.744 Pa

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Verified step by step guidance

First, understand that Young's modulus (E) is a measure of the stiffness of a material and is defined as the ratio of stress to strain. In this problem, we will use the relationship between the speed of sound in a material, its density, and Young's modulus.

The speed of sound (v) in a material is given by the formula: v = \( \sqrt{\frac{E}{\rho}} \), where E is Young's modulus and \( \rho \) is the density of the material.

We can rearrange this formula to solve for Young's modulus: E = \( v^2 \times \rho \).

Calculate the speed of sound in the metal bar using the formula: v = \( \frac{L}{t} \), where L is the length of the bar (1.5 m) and t is the time taken for sound to travel (3.9 × 10^-4 s).

Substitute the calculated speed of sound and the given density (6400 kg/m^3) into the rearranged formula for Young's modulus to find the value of E.

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Multiple Choice

A sound wave is emitted at a frequency of 300 Hz in air at a 0°C. As the sound wave travels through the air, the temperature increases. What is the wavelength of the sound wave at the following temperatures?

a. 0°C
b. 20°C
c. 45°C