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Physics for Scientists and Enginee...

10th Edition
Raymond A. Serway + 1 other
ISBN: 9781337553278

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Section
BuyFindarrow_forward

Physics for Scientists and Enginee...

10th Edition
Raymond A. Serway + 1 other
ISBN: 9781337553278
Textbook Problem

You are working for a plumber who is laying very long sections of copper pipe for a large building project. He spends a lot of time measuring the lengths of the sections with a measuring tape. You suggest a faster way to measure the length. You know that the speed of a one-dimensional compressional wave traveling along a copper pipe is 3.56 km/s. You suggest that a worker give a sharp hammer blow at one end of the pipe. Using an oscilloscope app on your smartphone, you will measure the time interval Δt between the arrival of the two sound waves due to the blow: one through the 20.0°C air and the other through the pipe. (a) To measure the length, you must derive an equation that relates the length L of the pipe numerically to the time interval Δt. (b) You measure a time interval of Δt = 127 ms between the arrivals of the pulses and, from this value, determine the length of the pipe. (c) Your smartphone app claims an accuracy of 1.0% in measuring time intervals. So you calculate by how many centimeters your calculation of the length might be in error.

(a)

To determine

The equation that relates the length L of the pipe numerically to the time interval.

Explanation

The speed of the compressional wave traveling along the pipe is 5.56km/s, length of the pipe is L and the time interval between the arrival of two wave at the other end, one through pipe and second through air is Δt.

Formula to calculate the time taken by wave to arrive at the other end through pipe is,

    L=vt1t1=Lv        (I)

Here, t1 is the time taken by wave to arrive through pipe and v is the speed of wave in pipe.

Formula to calculate the time taken by wave to arrive at the other end through air is,

    L=vst2t2=Lvs        (II)

Here, t2 is the time taken by wave to arrive through air and vs is the speed of wave in air.

Substitute equation (I) from equation (II).

    t2t1=LvsLvΔt=L(vvsv×vs)L=(v×vsvvs)Δt

Substitute 3.56km/s for v and 343m/s for vs in the above equation

(b)

To determine

The length of the pipe for Δt=127ms.

(c)

To determine

The error that might be occur in the length with 1% accuracy in the smartphone.

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