Concept explainers
Review. For the arrangement shown in Figure P14.60, the inclined plane and the small pulley are frictionless; the string supports the object of mass M at the bottom of the plane; and the string has mass m. The system is in equilibrium, and the vertical part of the string has a length h. We wish to study standing waves set up in the vertical section of the string. (a) What analysis model describes the object of mass M? (b) What analysis model describes the waves on the vertical part of the string? (c) Find the tension in the string. (d) Model the shape of the string as one leg and the hypotenuse of a right triangle. Find the whole length of the string. (e) Find the mass per unit length of the string. (f) Find the speed of waves on the string. (g) Find the lowest frequency for a standing wave on the vertical section of the string. (h) Evaluate this result for M = 1.50 kg, m = 0.750 g, h = 0.500 m, and θ = 30.0°. (i) Find the numerical value for the lowest frequency for a standing wave on the sloped section of the string.
Figure P14.60
(a)
The analysis model
Answer to Problem 60P
The object is described using constant acceleration model.
Explanation of Solution
The mass of the object supported by the string is
The object of mass
Conclusion:
Therefore, the object of mass
(b)
The analysis model
Answer to Problem 60P
The waves on the vertical part of the string can be described using the waves under boundary conditions model.
Explanation of Solution
The mass of the object supported by the string is
The vertical portion of the string is fixed at both the ends hence the boundary conditions stand for the waves. Write the general equation for the wavelength on the string.
Here,
Conclusion:
Therefore, the waves on the vertical part of the string can be described using the waves under boundary conditions model.
(c)
The tension on the string
Answer to Problem 60P
The tension on the string is
Explanation of Solution
Figure.1 shows the arrangement of the mass-pulley system.
Write the equation for the net force acting on the mass.
Here,
Conclusion:
Re-write the equation (I) such that the net force is zero.
Therefore, the tension on the string is
(d)
The length of the string
Answer to Problem 60P
The length of the string is
Explanation of Solution
Write the equation for the sine of the angle of inclination.
Rewrite the equation (I) to find the equation for the
Write the equation for the total length of the string.
Conclusion:
Substitute equation (II) in equation (III).
Therefore, the length of the strings is
(e)
The mass per unit length of the string
Answer to Problem 60P
The mass per unit length of the string is
Explanation of Solution
Write the equation for the mass per unit length of the string.
Here,
Conclusion:
Substitute equation (V) in equation (VI).
Therefore, the mass per unit length of the string is
(f)
The speed of waves on the string
Answer to Problem 60P
The speed of waves on the string is
Explanation of Solution
Write the equation for the speed of the waves on the string.
Here,
Conclusion:
Substitute equation (II) and equation (VIII) in equation (IX).
Therefore, the speed of the waves on the string is
(g)
The lowest frequency for a standing wave
Answer to Problem 60P
The lowest frequency for the standing waves is
Explanation of Solution
The length
Write the equation for the frequency of the standing wave.
Here,
Conclusion:
Substitute equation (X) and equation (XI) in equation (XII).
Therefore, the lowest possible frequency of the standing waves is
(h)
The lowest frequency for a set of given values
Answer to Problem 60P
The lowest frequency for the standing waves is
Explanation of Solution
Write the equation for the lowest frequency of the standing wave from equation (XIII).
Conclusion:
Substitute
Therefore, the lowest frequency of the standing waves is
(i)
The lowest frequency on the sloped section
Answer to Problem 60P
The lowest frequency for the standing waves on the sloped section is
Explanation of Solution
The wavelength of the fundamental mode of vibration is twice the length of the sloped section.
Here,
Substitute equation (X) and equation (XIV) in equation (XII).
Substitute
Therefore, the lowest frequency of the standing waves is
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Chapter 14 Solutions
Principles of Physics
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