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Chapter 13: Problem 45

A harmonic wave is traveling along a rope. It is observed that the oscillator that generates the wave completes \(40.0\) vibrations in \(30.0 \mathrm{~s}\). Also, a given maximum travels \(425 \mathrm{~cm}\) along the rope in \(10.0 \mathrm{~s}\). What is the wavelength?

Short Answer

Expert verified
The wavelength of the harmonic wave is approximately \(31.95 \mathrm{~cm}\).

Step by step solution

01

Find the frequency

Frequency (\(f\)) is the number of complete cycles per unit time. This would be found by dividing the number of vibrations by the total time, hence \(f = \frac{40.0}{30.0} = 1.33 \mathrm{~Hz}\).
02

Find the wave speed

Wave speed (\(v\)) is the distance covered by a point on the wave in a given time period. As we have been given the distance covered by a peak in a certain time, this is straightforward to calculate. Hence, \(v = \frac{425}{10.0} = 42.5 \mathrm{~cm/s}\). Note that the units of the velocity are in centimeters per second.
03

Find the Wavelength

To find the wavelength (\(\lambda\)), we can rearrange the wave speed equation to give \(\lambda = \frac{v}{f}\). Substituting the previously calculated values gives, \(\lambda = \frac{42.5}{1.33} = 31.95 \mathrm{~cm}\).

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Most popular questions from this chapter

A horizontal object-spring system oscillates with an amplitude of \(3.5 \mathrm{~cm}\) on a frictionless surface. If the spring constant is \(250 \mathrm{~N} / \mathrm{m}\) and the object has a mass of \(0.50 \mathrm{~kg}\), determine (a) the mechanical energy of the system, (b) the maximum speed of the object, and (c) the maximum acceleration of the object.

The distance between two successive minima of a transverse wave is \(2.76 \mathrm{~m}\). Five crests of the wave pass a given point along the direction of travel every \(14.0 \mathrm{~s}\). Find (a) the frequency of the wave and (b) the wave speed.

A \(326-g\) object is attached to a spring and executes simple harmonic motion with a period of \(0.250 \mathrm{~s}\). If the total energy of the system is \(5.83 \mathrm{~J}\), find (a) the maximum speed of the object, (b) the force constant of the spring, and (c) the amplitude of the motion.

A spring in a toy gun has a spring constant of \(9.80 \mathrm{~N} / \mathrm{m}\) and can be compressed \(20.0 \mathrm{~cm}\) beyond the equilibrium position. A \(1.00-\mathrm{g}\) pellet resting against the spring is propelled forward when the spring is released. (a) Find the muzzle speed of the pellet. (b) If the pellet is fired horizontally from a height of \(1.00 \mathrm{~m}\) above the floor, what is its range?

An object executes simple harmonic motion with an amplitude \(A\). (a) At what values of its position does its speed equal half its maximum speed? (b) At what values of its position does its potential energy equal half the total energy?
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