Ch. 15 - Wave Motion

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 15 - Wave Motion

Problem 75b

Chapter 15, Problem 75b

A bug on the surface of a pond is observed to move up and down a total vertical distance of 0.10 m, lowest to highest point, as a wave passes. If the amplitude increases to 0.15 m, by what factor does the bug’s maximum kinetic energy change?

Verified step by step guidance

Understand the relationship between the amplitude of a wave and the kinetic energy of an object oscillating due to the wave. The maximum kinetic energy of an oscillating object is proportional to the square of the amplitude of the wave. Mathematically, \( K_{max} \propto A^2 \), where \( K_{max} \) is the maximum kinetic energy and \( A \) is the amplitude.

Identify the initial amplitude \( A_1 \) and the final amplitude \( A_2 \). From the problem, \( A_1 = 0.10 \; \text{m} \) and \( A_2 = 0.15 \; \text{m} \).

Write the ratio of the maximum kinetic energies before and after the amplitude change. Using the proportionality \( K_{max} \propto A^2 \), the ratio is given by: \( \frac{K_{max,2}}{K_{max,1}} = \frac{A_2^2}{A_1^2} \).

Substitute the given values of \( A_1 \) and \( A_2 \) into the equation: \( \frac{K_{max,2}}{K_{max,1}} = \frac{(0.15)^2}{(0.10)^2} \).

Simplify the expression to find the factor by which the maximum kinetic energy changes. This will give the result for \( \frac{K_{max,2}}{K_{max,1}} \), which is the desired factor.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Amplitude

Amplitude is the maximum displacement of a wave from its equilibrium position. In the context of waves, it represents the height of the wave peaks and is directly related to the energy carried by the wave. A larger amplitude indicates that the wave has more energy, which affects the motion of objects, such as the bug in this scenario.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. In wave motion, the kinetic energy of an oscillating object, like the bug, is influenced by the amplitude of the wave, as greater amplitude leads to higher velocities at the maximum displacement points.

Energy Relationship in Waves

In wave mechanics, the energy of a wave is proportional to the square of its amplitude. This means that if the amplitude of a wave increases, the energy associated with the wave increases by the square of the factor of the amplitude change. Therefore, understanding this relationship is crucial for determining how changes in amplitude affect the kinetic energy of objects influenced by the wave.

Recommended video:

Guided course

03:43

Relationships Between Force, Field, Energy, Potential

Related Practice

Textbook Question

An earthquake-produced surface wave can be approximated by a sinusoidal transverse wave. Assuming a frequency of 0.60 Hz (typical of earthquakes, which actually include a mixture of frequencies), what amplitude is needed so that objects begin to leave contact with the ground? [Hint: Set the acceleration a > g. Why?]-

views

Textbook Question

Destructive interference occurs where two overlapping waves are 1/2 wavelength or 180° out of phase. Explain why 180° is equivalent to 1/2 wavelength.

views

Textbook Question

A guitar string is supposed to vibrate at 247 Hz, but is measured to actually vibrate at 262 Hz. By what percentage should the tension in the string be changed to get the frequency to the correct value?

views

Textbook Question

(II) For any type of wave that reaches a boundary beyond which its speed is increased, there is a maximum incident angle if there is to be a transmitted refracted wave. This maximum incident angle θ_iM corresponds to an angle of refraction equal to 90°. If θᵢ > θ_iM, all the wave is reflected at the boundary and none is refracted, because this would correspond to sin θᵣ > 1 (where is the angle θᵣ of refraction), which is impossible.

(a) Find a formula for θ_iM using the law of refraction, Eq. 15–19.

Textbook Question

A longitudinal earthquake wave strikes a boundary between two types of rock at a 41° angle. As the wave crosses the boundary, the specific gravity of the rock changes from 3.6 to 2.8. Assuming that the elastic modulus (Section 15–2)is the same for both types of rock, determine the angle of refraction.

views

Textbook Question

A transverse wave pulse travels to the right along a string with a speed v = 2.4 m/s. At t = 0 the shape of the pulse is given by the function D = 4.0m³ / (x² + 2.0m²), where D and x are in meters. Determine a formula for the wave pulse at any time t assuming the pulse is traveling to the left.

views