Fundamentals Of Physics - Volume 1 Only
Fundamentals Of Physics - Volume 1 Only
11th Edition
ISBN: 9781119306856
Author: Halliday
Publisher: WILEY
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Chapter 16, Problem 1Q
To determine

To rank:

The waves according to

a) their wave speed

b) the tension in the string along which they travel

Expert Solution & Answer
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Answer to Problem 1Q

Solution:

a) The waves can be ranked according to their wave speed as v1>v4>v2>v3(greatest first)

b) The waves can be ranked according to their tension in the string along which they travel as τ1>τ4>τ2>τ3(greatest first)

Explanation of Solution

1) Concept:

We can use the concept of the equation of transverse wave and speed of a travelling wave. The wave speed on a stretched string gives the relation between speed and tension in the string.

2) Formulae:

i) y=ym sinkx-ωt

ii)

v=ωk

iii)

v=τμ

3) Given:

The four waves along the strings with the same linear densities are

i) y1=3 mm sinx-3t

ii) y2=6 mm sin2x-t

iii) y3=1 mm sin4x-t

iv) y4=2 mm sinx-2t

4) Calculations:

a) Rank the waves according to their wave speed :

The equation of transverse wave is

y=ym sinkx-ωt(1)

The speed of the travelling wave is

v=ωk

The equation (i),

y1=3 mm sinx-3t

Compare this equation with equation (1), then the speed of the travelling wave is

v1=31

v1=3

The equation (ii) is

y2=6 mm sin2x-t

Compare this equation with equation (1), then the speed of the travelling wave is

v2=12

v2=0.5

The equation (iii) is

y3=1 mm sin4x-t

Compare this equation with equation (1), then the speed of the travelling wave is

v3=14

v3=0.25

The equation (iv) is

y4=2 mm sinx-2t

Compare this equation with equation (1), then the speed of the travelling wave is

v4=21

v4=2

Hence, the rank of the waves according to the wave speed is v1>v4>v2>v3 (greatest fitst).

b) Rank the waves according to tension:

The wave speed on a stretched string is

v=τμ

v ατ

The speed on the stretched string is directly proportional to the tension in the string with the same linear density.

The speed on the stretched string for equation (i) is

v1 ατ1

The speed on the stretched string for equation (ii) is

v2 ατ2

The speed on the stretched string for equation (iii) is

v3 ατ3

The speed on the stretched string for equation (i) is

v4 ατ4

Hence, the rank of the waves according to their tension is τ1>τ4>τ2>τ3 (greatest first).

Conclusion:

We can find the wave speed by using its expression and rank their values. By using the expression of the speed on the stretched string, we can find thetension in each string and rank their values.

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Chapter 16 Solutions

Fundamentals Of Physics - Volume 1 Only

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