Concept explainers
A spring of mass ms and spring constant k is attached to an object of mass M and set into
FIGURE P16.25
(a)
The kinetic energy of the system at the instant the object is moving with speed
Answer to Problem 25PQ
The kinetic energy of the system at the instant the object is moving with speed
Explanation of Solution
It is given that velocity of each segment
Write the expression for the kinetic energy of segment of mass
Here,
The velocity of segment is a function of distance from
This indicates that velocity of each segment varies linearly from
Write the expression for the velocity of segment at distance
Here,
Write the expression for mass of segment.
Here,
The kinetic energy of the system is the sum of kinetic energy of mass and total kinetic energy of the spring.
Write the expression for the total kinetic energy of system.
Here,
Write the expression for the kinetic energy of object.
Here,
Write the integral equation to find total kinetic energy of spring.
Substitute (II) and (III) in (VI) to get
Integrate above equation to get
Apply upper limit and lower limit to get
Substitute
Conclusion:
Therefore, the kinetic energy of the system at the instant the object is moving with speed
(b)
The frequency of oscillation of the system.
Answer to Problem 25PQ
The frequency of oscillation of the system is
Explanation of Solution
Write the expression for the frequency of oscillation.
Here,
Write the expression for
Here,
Substitute
Since mass is not located at end of the spring, the system must be considered to have spring constant and effective mass. Using total mass frequency of oscillation cannot be obtained.
Write expression for kinetic energy of system.
The kinetic energy is also equal that calculated in part (a).
Equate above equation with (VII) to get
Conclusion:
Substitute
Therefore, the frequency of oscillation of the system is
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Chapter 16 Solutions
Physics for Scientists and Engineers: Foundations and Connections
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