
Concept explainers
(a)
Then the distance ‘d’ to maximise the frequency of oscillation when a small initial displacement is given.

Answer to Problem 19.50P
Distance
Explanation of Solution
Given information:
Mass of collar
Mass of rod
Length of rod
The forces corresponding to the rod and collar are shown in the free body diagram below:
Now, from the equation of motion taking moment about A,
And, moment of inertia of rod is
Compare the above equation with un-damped equation of vibration:
Natural frequency:
To maximise the frequency, we need to take the derivative with respect to d and set it equal to zero.
By solving the above equation we get,
Conclusion:
The distance
(b)
The period of oscillation.

Answer to Problem 19.50P
Period of vibration,
Explanation of Solution
Given information:
Mass of collar
Mass of rod
Length of rod
The forces corresponding to the rod and collar are shown in the free body diagram below:
Now, from the equation of motion taking moment about A,
And, moment of inertia of rod is
Compare the above equation with un-damped equation of vibration;
Natural frequency:
To maximise the frequency, we need to take the derivative with respect to d and set it equal to zero.
By solving the above equation we get,
Thus,
Now, natural frequency:
Then, Time period
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Chapter 19 Solutions
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