Physics
Physics
3rd Edition
ISBN: 9780073512150
Author: Alan Giambattista, Betty Richardson, Robert C. Richardson Dr.
Publisher: McGraw-Hill Education
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Chapter 11, Problem 101P
To determine

The graphs of the transverse velocity and acceleration of the point x=0 as functions of time showing one complete cycle.

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Answer to Problem 101P

The graph of the transverse velocity of the point x=0 as function of time is

Physics, Chapter 11, Problem 101P , additional homework tip  1

And that of acceleration is

Physics, Chapter 11, Problem 101P , additional homework tip  2

Explanation of Solution

The given wave is harmonic.

Write the general expression for a harmonic wave.

y(x,t)=Asin(ωt+kx) (I)

Here, y(x,t) is the instantaneous displacement of the particles of the medium, A is the amplitude, ω is the angular velocity, t is the time, k is the wave vector and x is the distance travelled by the wave

Write the expression of the given standing wave.

y(x,t)=(2.00 mm)sin[(157 rad/s)t+(7.85 rad/m)x] (II)

Compare equations (I) and (II).

A=2.00 mmω=157 rad/sk=7.85 rad/m

Write the expression for the maximum velocity.

vm=ωA

Here, vm is the maximum velocity

Substitute 157 rad/s for ω and 2.00 mm for A in the above equation to find vm .

vm=(157 rad/s)(2.00 mm(1 m1000 mm))=(157 rad/s)(0.002 m)=0.314 m/s

Transverse velocity leads y(x,t) by a quarter cycle, so that it must be a cosine function.

Write the equation for transverse velocity.

vy=vmcos(ωt+kx)

Here, vy is the transverse velocity

Substitute 0.314 m/s for vm , 157 rad/s for ω and 2.00 mm for A in the above equation to find vy .

vy=(0.314 m/s)cos[(157 rad/s)t+(7.85 rad/m)x] (III)

Write the equation for the maximum acceleration for a point on the string.

am=ω2A

Here, am is the maximum acceleration

Substitute 157 rad/s for ω and 2.00 mm for A in the above equation to find am .

am=(157 rad/s)2(2.00 mm(1 m1000 mm))=(157 rad/s)2(0.002 m)=49.3 m/s2

Transverse acceleration leads vy by a quarter cycle so that it leads y(x,t) by a half cycle. This implies transverse acceleration is a negative sine function.

Write the equation for the transverse acceleration.

ay=amsin(ωt+kx)

Here, ay is the transverse acceleration

Substitute 49.3 m/s2 for am , 157 rad/s for ω and 2.00 mm for A in the above equation to find ay .

ay=(49.3 m/s2)sin[(157 rad/s)t+(7.85 rad/m)x] (IV)

Write the equation for the time period of the wave.

T=2πω

Here, T is the time period of the wave

Substitute 157 rad/s for ω in the above equation to find T .

T=2π157 rad/s=40.0×103 s=40.0 ms

Conclusion:

Substitute 0 for x in equation (III).

vy=(0.314 m/s)cos[(157 rad/s)t+(7.85 rad/m)0]=(0.314 m/s)cos[(157 rad/s)t]

The graph of the above equation is plotted in figure 1.

Physics, Chapter 11, Problem 101P , additional homework tip  3

Substitute 0 for x in equation (IV).

ay=(49.3 m/s2)sin[(157 rad/s)t+(7.85 rad/m)0]=(49.3 m/s2)sin[(157 rad/s)t]

The graph of the above equation is plotted in figure 2.

Physics, Chapter 11, Problem 101P , additional homework tip  4

Thus, the graphs of the transverse velocity and acceleration of the point x=0 as functions of time showing one complete cycle are plotted.

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