Concept explainers
(a)
The force constant of the spring.
(a)
Answer to Problem 28P
The force constant of the spring is
Explanation of Solution
Given that the force is
Since the horizontal force of
Write the expression for the force constant of the spring.
Here,
Conclusion:
Substitute
Therefore, the force constant of the spring is
(b)
The frequency of oscillation of the spring-object system.
(b)
Answer to Problem 28P
The frequency of oscillation of the spring-object system is
Explanation of Solution
Given that the mass of the object is
Write the expression for the frequency of oscillation of the spring-object system.
Here,
Conclusion:
Substitute
Therefore, the frequency of oscillation of the spring-object system is
(c)
The maximum speed of the object.
(c)
Answer to Problem 28P
The maximum speed of the object is
Explanation of Solution
Given that the displacement from the equilibrium position is
Write the expression for the maximum speed of the object executing
Here,
The amplitude of motion is equal to the initial displacement of the object which is
Write the expression for the angular frequency.
Use equation (IV) in (III).
Conclusion:
Substitute
Therefore, the maximum speed of the object is
(d)
The position where maximum speed occur.
(d)
Answer to Problem 28P
The position where maximum speed occur is
Explanation of Solution
For the oscillating spring-object system, the objects loses all its potential energy and gains maximum kinetic energy at the equilibrium point. Since the maximum kinetic energy corresponds to the maximum speed, the object gains maximum speed at the equilibrium position, which is described by the coordinate,
Conclusion:
Therefore, the position where maximum speed occur is
(e)
The maximum acceleration of the object.
(e)
Answer to Problem 28P
The maximum acceleration of the object is
Explanation of Solution
Write the expression for the maximum acceleration of the object executing SHM.
Here,
Use equation (IV) in (VI).
Conclusion:
Substitute
Therefore, the maximum acceleration of the object is
(f)
The position where the maximum acceleration occur.
(f)
Answer to Problem 28P
The position where the maximum acceleration occur is
Explanation of Solution
For the oscillating spring-object system, the maximum acceleration occurs where the object reverses its direction of motion. This happens only at the positions corresponding to the maximum displacement. Which is
Conclusion:
Therefore, the position where the maximum acceleration occur is
(g)
The total energy of the oscillating system.
(g)
Answer to Problem 28P
The total energy of the oscillating system is
Explanation of Solution
It is obtained that the force constant of the spring is
Write the expression for the energy of the spring-object oscillating system.
Here,
Conclusion:
Substitute
Therefore, the total energy of the oscillating system is
(h)
The speed of the object when its position is equal to one-third the maximum value.
(h)
Answer to Problem 28P
The speed of the object when its position is equal to one-third the maximum value is
Explanation of Solution
Write the expression for the speed at a given position of an object executing SHM in a spring.
Here,
Since the position is one-third the maximum value (
Conclusion:
Substitute
Therefore, the speed of the object when its position is equal to one-third the maximum value is
(i)
The acceleration of the object when its position is equal to one-third the maximum value.
(i)
Answer to Problem 28P
The speed of the object when its position is equal to one-third the maximum value is
Explanation of Solution
Write the expression for the acceleration at a given position of an object executing SHM in a spring.
Here,
Since the position is one-third the maximum value (
Conclusion:
Substitute
Therefore, the speed of the object when its position is equal to one-third the maximum value is
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Chapter 15 Solutions
Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University
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