Concept explainers
A 2.00-kg object is attached to a spring and placed on a frictionless, horizontal surface. A horizontal force of 20.0 N is required to hold the object at rest when it is pulled 0.200 m from its equilibrium position (the origin of the x axis). The object is now released from rest from this stretched position, and it subsequently undergoes simple harmonic oscillations. Find (a) the force constant of the spring, (b) the frequency of the oscillations, and (c) the maximum speed of the object, (d) Where does this maximum speed occur? (e) Find the maximum acceleration of the object. (f) Where does the maximum acceleration occur? (g) Find the total energy of the oscillating system. Find (h) the speed and (i) the acceleration of the object when its position is equal to one-third the maximum value.
(a)
The force constant of the spring.
Answer to Problem 15.28P
The force constant of the spring is
Explanation of Solution
Given info: The force required to hold the object at rest is
Write the expression for force constant.
Here,
Substitute
Conclusion:
Therefore, the force constant of the spring is
(b)
The frequency of the oscillations.
Answer to Problem 15.28P
The frequency of the oscillations is
Explanation of Solution
Given info: The force required to hold the object at rest is
Write the expression for the force constant of the spring.
Here,
Substitute
Write the expression for the frequency of the oscillations.
Here,
Substitute
Conclusion:
Therefore, the frequency of the oscillations is
(c)
The maximum speed of the object.
Answer to Problem 15.28P
The maximum speed of the object is
Explanation of Solution
Given info: The force required to hold the object at rest is
From part (b) the angular velocity is
Write the expression for maximum speed.
Here,
Substitute
Conclusion:
Therefore, the maximum speed of the object is
(d)
The position of the object where the maximum speed occurs.
Answer to Problem 15.28P
The position of the object where the maximum speed occurs at
Explanation of Solution
Given info: The force required to hold the object at rest is
The maximum speed of the object occurs when the object passes through its equilibrium position.
The equilibrium position of the object is,
Conclusion:
Therefore, the position of the object where the maximum speed occurs at
(e)
The maximum acceleration of the object.
Answer to Problem 15.28P
The maximum acceleration of the object is
Explanation of Solution
Given info: The force required to hold the object at rest is
Write the expression for the maximum acceleration.
Here,
Substitute
Conclusion:
Therefore, the maximum acceleration of the object is
(f)
The position of the object where the maximum acceleration occurs.
Answer to Problem 15.28P
The position of the object where the maximum acceleration occurs at
Explanation of Solution
Given info: The force required to hold the object at rest is
The maximum acceleration of the object occurs when the object reverses its direction of motion.
The object reverses its direction of motion where its distance is the maximum from the equilibrium position.
The maximum distance from equilibrium position of the object is,
Conclusion:
Therefore, the position of the object where the maximum acceleration occurs at
(g)
The total energy of the system.
Answer to Problem 15.28P
The total energy of the system is
Explanation of Solution
Given info: The force required to hold the object at rest is
From part (a) the force constant of the spring is
Write the expression for the total energy.
Here,
Substitute
Conclusion:
Therefore, the total energy of the system is
(h)
The speed of the object when the object is at one-third of the maximum value.
Answer to Problem 15.28P
The speed of the object when the object is at one-third of the maximum value is
Explanation of Solution
Given info: The force required to hold the object at rest is
The position of the object is one-third of the maximum value.
Here,
Substitute
Thus, the value of
From part (a) the angular velocity is
Write the expression for the velocity of block at any position.
Here,
Substitute
Conclusion:
Therefore, the speed of the object when the object is at one-third of the maximum value is
(i)
The acceleration of the object when the object is at one-third of the maximum value.
Answer to Problem 15.28P
The acceleration of the object when the object is at one-third of the maximum value is
Explanation of Solution
Given info: The force required to hold the object at rest is
From part (h) the position of the object is one-third of the maximum value is
From part (a) the angular velocity is
Write the expression for the acceleration of block at any position.
Here,
Substitute
Conclusion:
Therefore, the acceleration of the object when the object is at one-third of the maximum value is
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Chapter 15 Solutions
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