Concept explainers
As in Figure P18.16, a simple harmonic oscillator is attached to a rope of linear mass density 5.4 × 10−2 kg/m, creating a standing transverse wave. There is a 3.6-kg block hanging from the other end of the rope over a pulley. The oscillator has an angular frequency of 43.2 rad/s and an amplitude of 24.6 cm. a. What is the distance between adjacent nodes? b. If the angular frequency of the oscillator doubles, what happens to the distance between adjacent nodes? c. If the mass of the block is doubled instead, what happens to the distance between adjacent nodes? d. If the amplitude of the oscillator is doubled, what happens to the distance between adjacent nodes?
FIGURE P18.16
(a)
The distance between the adjacent nodes.
Answer to Problem 16PQ
The distance between the adjacent nodes is
Explanation of Solution
Given that the linear mass density of the rope is
Write the expression for the wavelength of the wave.
Here,
Write the expression for the speed of the wave.
Here,
Write the equation to find the frequency of the wave.
Here,
Use equation (III) and (II) in (I).
Write the expression for the tension force on the rope (it is equal to the weight of the block hanged).
Here,
Use above expression in equation (IV).
The distance between the adjacent nodes is equal to half the wavelength of the wave.
Rewrite above equation using equation (V).
Conclusion:
Substitute
Therefore, the distance between the adjacent nodes is
(b)
The distance between the adjacent nodes when the angular frequency is doubled.
Answer to Problem 16PQ
The distance between the adjacent nodes when the angular frequency is doubled is
Explanation of Solution
Equation (VI) gives the expression for the distance between the nodes.
From the above equation it is clear that wavelength is inversely proportional to angular frequency.
The angular frequency is doubled. Replace
Conclusion:
Substitute
Therefore, distance between the adjacent nodes when the angular frequency is doubled is
(c)
The distance between the adjacent nodes if the mass of the block is doubled.
Answer to Problem 16PQ
The distance between the adjacent nodes if the mass of the block is doubled is
Explanation of Solution
Equation (VI) gives the expression for the distance between the nodes.
When the mass of the block is doubled,
Conclusion:
Substitute
Therefore, the distance between the adjacent nodes if the mass of the block is doubled is
(d)
The distance between the nodes if the amplitude of the oscillator is doubled.
Answer to Problem 16PQ
The distance between the nodes remains the same even if the amplitude of the oscillator is doubled.
Explanation of Solution
Equation (VI) gives the expression for the distance between the nodes.
The above equation is independent of the amplitude term. Thus, even if the amplitude of the oscillator is doubled, the distance between the nodes do not change.
Conclusion:
Therefore, the distance between the nodes remains the same even if the amplitude of the oscillator is doubled.
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Chapter 18 Solutions
Physics for Scientists and Engineers: Foundations and Connections
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