Question

12.

A longitudinal standing wave can be created in a long, thin aluminum rod by stroking the rod with very dry fingers. This is often done as a physics demonstration, creating a high-pitched, very annoying whine. From a wave perspective, the standing wave is equivalent to a sound standing wave in an open-open tube. In particular, both ends of the rod are anti-nodes.

What is the fundamental frequency of a 2.50 m -long aluminum rod? The speed of sound in aluminum is 6420 m/s.

Express your answer with the appropriate units.

7.

The lowest note on a grand piano has a frequency of 27.5 Hz. The entire string is 2.00 m long and has a mass of 470 g . The vibrating section of the string is 1.59 m long.

What tension is needed to tune this string properly?

Express your answer with the appropriate units.

6.

A 2.5 m -long string is fixed at both ends and tightened until the wave speed is 30 m/s .

What is the frequency of the standing wave shown in the figure (Figure 1)?

Express your answer to two significant figures and include the appropriate units.

3.

Learning Goal:

To understand the concept of nodes of a standing wave.

The nodes of a standing wave are points where the displacement of the wave is zero at all times. Nodes are important for matching boundary conditions, for example that the point at which a string is tied to a support has zero displacement at all times (i.e., the point of attachment does not move).

Consider a standing wave, where y represents the transverse displacement of a string that extends along the x direction. Here is a common mathematical form for such a wave:

y(x,t)=Acos(kx)sin(ωt),

where A is the maximum transverse displacement of the string (the amplitude of the wave), which is assumed to be nonzero, k is the wavenumber, ω is the angular frequency of the wave, and t is time.

What is the displacement of the string as a function of x at time T/4, where T is the period of oscillation of the string?

Express the displacement in terms of A, x, and k only. That is, evaluate ω⋅T4 and substitute it in the equation for y(x,t).

2.

Learning Goal:

To understand the concept of normal modes of oscillation and to derive some properties of normal modes of waves on a string.

A normal mode of a closed system is an oscillation of the system in which all parts oscillate at a single frequency. In general there are an infinite number of such modes, each one with a distinctive frequency fi and associated pattern of oscillation.

Consider an example of a system with normal modes: a string of length L held fixed at both ends, located at x=0 and x=L. Assume that waves on this string propagate with speed v. The string extends in the x direction, and the waves are transverse with displacement along the y direction.

In this problem, you will investigate the shape of the normal modes and then their frequency.

The normal modes of this system are products of trigonometric functions. (For linear systems, the time dependance of a normal mode is always sinusoidal, but the spatial dependence need not be.) Specifically, for this system a normal mode is described by

yi(x,t)=Aisin(2πxλi)sin(2πfit).

Find the three longest wavelengths (call them λ1, λ2, and λ3) that "fit" on the string, that is, those that satisfy the boundary conditions at x=0 and x=L. These longest wavelengths have the lowest frequencies.

Express the three wavelengths in terms of L. List them in decreasing order of length, separated by commas.

14.

Piano tuners tune pianos by listening to the beats between the harmonics of two different strings. When properly tuned, the note A should have a frequency of 440 Hz and the note E should be at 659 Hz.

A tuner first tunes the A string very precisely by matching it to a 440 Hz tuning fork. She then strikes the A and E strings simultaneously and listens for beats between the harmonics. What beat frequency between higher harmonics indicates that the E string is properly tuned?

Express your answer with the appropriate units.

Answer 1

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