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-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 $\mu$g. You then hang a 0.400-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 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 In this example, we're determining the length of a long wire that's suspended from the ceiling.

00:07 We're going to be finding the length by measuring the wave speed of a wave pulse, okay, from one end of the wire to the other.

00:17 Okay, so we have this wire suspended from the ceiling.

00:22 Okay, we can't measure the length directly.

00:25 We don't know the mass, but what we do have is a small sample of the wire, okay, and this sample has a length l of 2 .00 centimeters and it has a mass m of 14 .5 micrograms.

00:48 Okay, we are going to hang a mass from the bottom of this wire.

00:54 That mass is going to be m, capital m, equals 0 .400 kilograms.

01:05 Okay, and we have a sensor at the top of the wire, so at the ceiling and here at the bottom, to measure the waves, the time it takes for the wave to travel on the full length of the wire, so from top to bottom or bottom to top, and that time is t equals 26 .7 milliseconds.

01:31 Okay, so using this information, we can determine the length of the wire and how do we do that? well, we have an equation for the wave speed for a wave traveling in a wire as a function of the tension in the wire or string and the linear mass density, okay, and that is given by velocity equals the square root of tension divided by the linear mass density.

02:03 Okay, so let's find our linear mass density and then we'll have velocity.

02:08 Okay, then we can use our relationship between velocity, time, and distance to find the total length and i'll call this l of the wire.

02:19 Okay, so let's determine the linear mass density, mu, and that's simply mass over length.

02:32 Okay, so we have 14 micrograms, 14 .5 divided by 2 centimeters, so we have 7 .25 micrograms per centimeter.

02:51 Okay, let's put that in terms of kilograms and meters...

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