Concept explainers
A small collar of mass 1 kg is rigidly attached to a 3-kg uniform rod of length L = 750 mm. Determine (a) the distance d to maximize the frequency of oscillation when the rod is given a small initial displacement, (b) the corresponding period of oscillation.
Fig. P19.50
(a)
The distance d to maximize the frequency of oscillation when the rod is given a small initial displacement.
Answer to Problem 19.50P
The distance d to maximize the frequency of oscillation when the rod is given a small initial displacement is
Explanation of Solution
Given information:
The mass
The mass
The length (L) of the rod AB is 750 mm.
The acceleration due to gravity (g) is
Calculation:
Show the free-body-diagram equation as in Figure (1).
The external forces in the system are forces due mass of the collar and the rod. The effective force in the system is
Take moment about A in the system for external forces.
For small oscillation
Take moment about A in the system for effective forces.
The tangential component of acceleration for the rod
The tangential component of acceleration for the collar
Thus, express the moment about A due to the effective forces as:
Equate the moment about A in the system for external and effective forces.
Compare the differential Equation (1) with the general differential equation of motion
Write the expression for moment of inertia of the rod:
Substitute 1 kg for
Calculate the value of d to maximize the natural frequency:
Differentiate Equation (3) with respect to d.
Equate
Solve the above quadratic equation:
Express the roots of a quadratic equation:
Substitute d for x, 1 for a, 2.25 for b, and -0.5625 for c to find the roots of the Equation (4).
Therefore, the distance d to maximize the frequency of oscillation when the rod is given a small initial displacement is
(b)
The corresponding period
Answer to Problem 19.50P
The corresponding period
Explanation of Solution
Given information:
The mass
The mass
The length (L) of the rod AB is 750 mm.
The acceleration due to gravity (g) is
Calculation:
Calculate the value of maximum natural circular frequency
Substitute 0.227 m for d in Equation (3)
Calculate the time period of oscillation
Substitute
Therefore, the corresponding period
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Chapter 19 Solutions
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