Concept explainers
(a)
The position of
(a)
Answer to Problem 46P
The position of
Explanation of Solution
The general expression for the position of the object is given by,
Here,
Write the expression for the force due to a spring.
Here,
Write the expression for the force acting on an object.
Here,
Equate equation (II) and (III) and solve for
Take the
Write the expression for the angular frequency.
Conclusion:
Substitute
Substitute
Substitute
Therefore, The position of
(b)
The distance traveled by the vibrating object in part (a).
(b)
Answer to Problem 46P
The distance traveled by the vibrating object in part (a) is
Explanation of Solution
The distance traveled is equal to the number of oscillations multiplied by the distance traveled in a single time period.
The time period is given by,
Here,
The number of oscillations made by the spring is given by,
Here,
The distance travelled by the mass in one complete oscillation is
The total distance traveled in
Conclusion:
Substitute
Substitute
Substitute e
Therefore, The distance traveled by the vibrating object in part (a) is
(c)
The position of the object
(c)
Answer to Problem 46P
The position of the object
Explanation of Solution
Substitute
Substitute
Substitute
Therefore, the position of
Conclusion:
Therefore, the position of the object
(d)
The distance traveled by the object in part (c)
(d)
Answer to Problem 46P
The distance traveled by the object in part (c) is
Explanation of Solution
The distance traveled is equal to the number of oscillations multiplied by the distance traveled in a single time period.
The time period is given by,
The number of oscillations made by the spring is given by,
The distance travelled by the mass in one complete oscillation is
The total distance traveled in
Conclusion:
Substitute
Substitute
Substitute e
Therefore, the distance traveled by the vibrating object in part (c) is
(e)
The reason for the different answers to part (a) and (c) when the initial data in parts (a) and (c) are so similar and the answers to parts (b) and (d) are relatively close.
(e)
Answer to Problem 46P
The answers in parts a and b are different because of the difference in angular velocities.
Explanation of Solution
Diverging patterns of oscillations which starts out in phase but becoming completely out of phase changes the answers in part (a) and (b). Since the
For parts c and d, the distance traveled depends only on the angular velocity such that their difference is not so large.
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Chapter 12 Solutions
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