Concept explainers
(a) A hanging spring stretches by 35.0 cm when an object of mass 450 g is hung on it at rest. In this situation, we define its position as x = 0. The object is pulled down an additional 18.0 cm and released from rest to oscillate without friction. What is its position x at a moment 84.4 s later? (b) Find the distance traveled by the vibrating object in part (a), (c) What If? Another hanging spring stretches by 35.5 cm when an object of mass 440 g is hung on it at rest. We define this new position as x = 0. This object is also pulled down an additional 18.0 cm and released from rest to oscillate without friction. Find its position 84.4 s later, (d) Find the distance traveled by the object in part (c). (e) Why are the answers to parts (a) and (c) so different when the initial data in parts (a) and (c) are so similar and the answers to parts (b) and (d) are relatively close? Does this circumstance reveal a fundamental difficulty in calculating the future?
(a)
The position
Answer to Problem 15.12P
The position
Explanation of Solution
Given info: The distance through which spring stretches is
Write the equation of position of an object attached to a spring.
Here,
Write the equation for angular frequency.
Here,
Write the equation of force generated due to stretch in spring.
Here,
Write the equation of force due to the weight of the object.
Here,
Substitute
Substitute
Substitute
Substitute
Conclusion:
Therefore, the position
(b)
The distance travelled by the object in part (a).
Answer to Problem 15.12P
The distance travelled by the object is
Explanation of Solution
Given info: The distance through which spring stretches is
Write the equation of distance travelled by the object for complete oscillation.
Here,
The total distance
Substitute
The number of complete oscillation for angular displacement of
Substitute
Conclusion:
Therefore, the distance travelled by the object is
(c)
The position of the object.
Answer to Problem 15.12P
The position
Explanation of Solution
Given info: The distance through which spring stretches is
Substitute
Conclusion:
Therefore, the position
(d)
The distance travelled by the object in part (c).
Answer to Problem 15.12P
The distance travelled by the object is
Explanation of Solution
Given info: The distance through which spring stretches is
Write the equation of distance travelled by the object for complete oscillation.
The number of complete oscillation for angular displacement of
Substitute
Conclusion:
Therefore, the distance travelled by the object is
(e)
The reason that the answer of parts (a) and (c) is different when the initial data in parts (a) and (c) are similar and the answers of parts (b) and (d) are relatively close and whether this circumstance give the fundamental difficulty.
Answer to Problem 15.12P
The oscillation patterns diverge from each other, starts out in phase and becomes out of phase completely. It is impossible to make future predictions with the known data.
Explanation of Solution
Given info: The distance through which spring stretches is
The pattern of oscillation diverges completely from each other even if the initial data are same. Starts out in phase initially, but becomes completely out of phase.
To calculate the future predictions, exact data of the present is required which is impossible. It is difficult to make prediction with the known data.
Thus, the oscillation patterns diverge from each other, starts out in phase and becomes out of phase completely. It is impossible to make future predictions with the known data.
Conclusion:
Therefore, the oscillation patterns diverge from each other, starts out in phase and becomes out of phase completely. It is impossible to make future predictions with the known data.
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