Concept explainers
(a)
The first two harmonic frequencies of vibration for the wire.
(a)
Answer to Problem 32PQ
The first two harmonic frequencies of vibration for the wire are
Explanation of Solution
Write the expression for the natural frequencies of vibration of a wire fixed at both ends.
Here,
Write the expression for the speed of wave in terms of tension in string and mass per unit length.
Here,
Write the equation for
Here,
Put the above equation in equation (II).
Put equation (III) in (I) to get final expression for
Conclusion:
It is given that length of the wire is
The first two harmonics are obtained when
Substitute
Substitute
Therefore, the first two harmonic frequencies of vibration for the wire are
(b)
The possible harmonic modes of vibration of the wire.
(b)
Answer to Problem 32PQ
The possible harmonic modes of vibration of the wire are
Explanation of Solution
It is given that the wire has a mode at
Write the expression for
Here,
Rewrite the above equation for
Write the expression for the wavelength of harmonics.
Rewrite the above equation for
Put equation (V) in equation (VI).
Substitute
Thus the mode corresponds to the
Substitute
There could be also an entire wavelength fitting between the nodes at the end at
Write the expression for
Here,
Rearrange the above equation for
Put the above equation in equation (VI).
Substitute
The expression for
Conclusion:
Substitute
Substitute
Therefore, the possible harmonic modes of vibration of the wire are
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Chapter 18 Solutions
Physics for Scientists and Engineers: Foundations and Connections