You must determine the length of a long, thin wire that is...

You must determine the length of a long, thin wire that is suspended from the ceiling in the atrium of a tall building. A $2.00-\mathrm{cm}$ -long piece of the wire is left over from its installation. Using an analytical balance, you determine that the mass of the spare piece is $14.5 \mu \mathrm{g}$. You then hang a $0.400 \mathrm{~kg}$ mass from the lower end of the long, suspended wire. When a small-amplitude transverse wave pulse is sent up that wire, sensors at both ends measure that it takes the wave pulse $26.7 \mathrm{~ms}$ to travel the length of the wire. (a) Use these measurements to calculate the length of the wire. Assume that the weight of the wire has a negligible effect on the speed of the transverse waves. (b) Discuss the accuracy of the approximation made in part (a).

Key Concepts

Linear Mass Density

This concept involves determining the mass per unit length of the wire by dividing the mass of a small, measurable piece by its length. It is a fundamental property for extended bodies and is used to relate the mass and dimensions in calculations involving wave speed and dynamics along a continuous medium.

Transverse Wave Propagation on a String

Transverse waves are disturbances that travel perpendicular to the direction of the medium. In the context of a wire or string, the characteristics of these waves (including their speed) depend on the tension in the wire and its linear mass density, determining how disturbances travel along the medium.

Wave Speed and Tension Relationship

The speed of a transverse wave on a string is given by the formula v = sqrt(T/?), where T is the tension and ? is the linear mass density. This relationship is crucial in linking the mechanical properties of the wire with the observed wave behavior, allowing for determination of unknown quantities, such as the length of the wire via measured travel times.

Time-of-Flight Measurement

This concept refers to measuring the time it takes a wave pulse to travel along the length of the wire. When combined with the calculated wave speed from the tension and mass per unit length, this time measurement can be used to determine the total length of the wire by the simple relation L = v*t.

Assumptions and Approximations in Wave Propagation

The analysis assumes that the weight or mass of the wire has a negligible effect on the tension along its length, meaning the tension is considered uniform. This simplification is common in introductory treatments, though in real-world scenarios the variable tension due to the wire's own weight could affect wave speed and the accuracy of length determination.

Transcript

00:01 We have this very long suspended from the ceiling, and we want to find its length, given some information about its wave properties and about its weight, etcetera.

00:10 So we have a sample of this wire.

00:12 Ah, basically, that's two centimetres long.

00:15 Um, and it's got a certain mass of 14.5 micrograms.

00:19 So let's calculate the mass per unit length of the wire using this information so we know it's 14.5 micro.

00:27 So my gross times 10 to the minus six grams, which is times 10 to the minus three kilograms.

00:35 So it's mass breeding length master length, about two centimeters, 10 to the minus two meters, and that's gonna be equal to 7.25 times.

00:54 This is 10 to the minus nine.

00:56 Might is 10 to the minus two, turning them on a seven meters here.

01:01 So we have the mass per unit length, and we're also told that if you send a wave pulse up that it takes 26.7 milliseconds.

01:12 So called that delta t to send a wave pulse up is 26 0.7 milliseconds, and we want to know the length.

01:24 So let's take let's figure out what we know about lincoln.

01:28 Time and velocity.

01:30 Ah, first, and we know that the velocity of a wave is going to be equal.

01:37 Thio.

01:38 Actually, let's say the length of this is going to be the velocity of the wave times the time it takes for a wave pulse to travel up, says the times.

01:45 26.7 times 10 to the minus three seconds.

01:54 So we can solve for v and say that the velocity is going to be equal to l over delta t, which is this year.

02:03 But we also know and the reason why i picked out v hopefully should be clear to you as a variable to focus on because we know this relationship v is the square root of attention over the mass per unit like and the problem says we're supposed to make this assumption...

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