Physics for Scientists and Engineers
Physics for Scientists and Engineers
6th Edition
ISBN: 9781429281843
Author: Tipler
Publisher: MAC HIGHER
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Chapter 16, Problem 68P

(a)

To determine

Frequencies of the lowest four harmonics if string is fixed at both ends.

(a)

Expert Solution
Check Mark

Explanation of Solution

Given:

Length of string is 35m .

Linear mass density is 0.0085kg/m .

Tension in string is 18N .

Formula used:

Write expression for nth frequency of the transverse wave.

  fn=vλn

Substitute 2Ln for λn in above expression.

  fn=v 2Lnfn=nv2L

Substitute Tμ for v in above expression.

  fn=n2LTμ......... (1)

Calculation:

Substitute 1 for n , 35m for L , 18N for T and 0.0085kg/m for μ in equation (1).

  f1=12( 35m) 18N 0.0085 kg/m f1=0.657Hz

Substitute 2 for n , 35m for L , 18N for T and 0.0085kg/m for μ in equation (1).

  f2=22( 35m) 18N 0.0085 kg/m f2=1.31Hz

Substitute 3 for n , 35m for L , 18N for T and 0.0085kg/m for μ in equation (1).

  f3=32( 35m) 18N 0.0085 kg/m f3=1.97Hz

Substitute 4 for n , 35m for L , 18N for T and 0.0085kg/m for μ in equation (1).

  f4=42( 35m) 18N 0.0085 kg/m f4=2.63Hz

Conclusion:

Thus, frequencies of the lowest four harmonics are 0.657Hz , 1.31Hz , 1.97Hz and 2.63Hz .

(b)

To determine

Frequencies of the lowest four harmonics if string is fixed at one end.

(b)

Expert Solution
Check Mark

Explanation of Solution

Given:

Length of string is 35m .

Linear mass density is 0.0085kg/m .

Tension in string is 18N .

Formula used:

Write expression for nth frequency of the transverse wave.

  fn=vλn

Substitute 4Ln for λn in above expression.

  fn=v 4Lnfn=nv4L

Substitute Tμ for v in above expression.

  fn=n4LTμ......... (1)

Calculation:

Substitute 1 for n , 35m for L , 18N for T and 0.0085kg/m for μ in equation (1).

  f1=14( 35m) 18N 0.0085 kg/m f1=0.329Hz

Substitute 3 for n , 35m for L , 18N for T and 0.0085kg/m for μ in equation (1).

  f3=34( 35m) 18N 0.0085 kg/m f3=0.987Hz

Substitute 5 for n , 35m for L , 18N for T and 0.0085kg/m for μ in equation (1).

  f5=54( 35m) 18N 0.0085 kg/m f5=1.65Hz

Substitute 4 for n , 35m for L , 18N for T and 0.0085kg/m for μ in equation (1).

  f7=74( 35m) 18N 0.0085 kg/m f7=2.30Hz

Conclusion:

Thus, frequencies of the lowest four harmonics are 0.329Hz , 0.987Hz , 1.65Hz and 2.30Hz .

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