Concept explainers
(a)
The expression for the radius of the sphere in the water.
(a)
Answer to Problem 73AP
The expression for the radius of the sphere in the water is
Explanation of Solution
Write the expression for tension in the string (Refer Figure 18.11a).
Here,
Write the expression for tension on the string included the buoyant force on the sphere (Refer Figure 18.11b).
Here,
Write the expression for the buoyant force acts on the sphere.
Here,
Write the expression for volume of the sphere.
Here,
Write the expression for the frequency of the oscillation.
Here,
Write the expression for the fundamental frequency of the oscillation.
Write the expression for frequency of the two antinodes formed on the string.
Conclusion:
Substitute the equation (III) and (IV) in equation (II).
Rewrite the equation (V) and (VI).
Substitute equation (I) in the above equation.
Substitute equation (VII) in the above equation.
Solve the above relation for radius.
Substitute
Therefore, the expression for the radius of the sphere in the water is
(b)
The minimum allowed value of n.
(b)
Answer to Problem 73AP
The minimum allowed value of n is
Explanation of Solution
The factor inside the cubic root is,
Conclusion:
Since the above factor will be either zero or negative which are meaningless results, for
Therefore, the minimum allowed value of n is
(c)
The radius of the largest sphere producing a standing wave on the string.
(c)
Answer to Problem 73AP
The radius of the largest sphere producing a standing wave on the string is
Explanation of Solution
The mass of the sphere is held constant while its radius is changed, there will reach a point where the density of the sphere reaches the density of the water, and then the sphere will float on the water.
Write the expression for the density of the sphere.
Here,
Rearrange the above solution for r.
Conclusion:
Substitute
Therefore, the radius of the largest sphere producing a standing wave on the string is
(d)
Can larger sphere is used, what will happen.
(d)
Answer to Problem 73AP
The sphere floats on the water.
Explanation of Solution
The mass of the sphere is held constant while its radius is changed, it will reach a point where the density of the sphere reaches the density of the water, and then the sphere will float on the water.
Conclusion:
If the large sphere is used, then the sphere will float on the water.
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Chapter 18 Solutions
Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University
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