Concept explainers
(a)
Find the amplitude of the motion
(a)
Answer to Problem 19.110P
The amplitude of the motion
Explanation of Solution
Given information:
The weight of the bob
The weight of the collar
The length of the simple pendulum (l) is
The amplitude of the collar
The magnitude of static deflection
The frequency
The acceleration due to gravity (g) is
Calculation:
Calculate the mass of the bob
Substitute
Calculate the mass of the collar
Substitute
Calculate the frequency of the period
Substitute
Show the system at before and after moving the collar horizontally as in Figure 1.
Refer Figure (1),
Before giving horizontal movement, the force mg is equal to the tension in the pendulum.
The expression for the force balance equation for initial condition as follows:
Calculate the value of
The expression for the force balance equation in x-direction in the displaced condition as follows:
Here,
The only force in the x-direction is the tension component
Substitute
Calculate the differential equation of motion using the formula:
Substitute mg for T and
Substitute
Here,
Compare the differential equation (2) with the general differential equation of motion for forced vibration
Calculate the natural circular frequency of vibration
Substitute
Calculate the amplitude of the forced vibration
Substitute
Therefore, the amplitude of the motion
(b)
Find the force that must be applied to collar C (F) to maintain the motion.
(b)
Answer to Problem 19.110P
The force that must be applied to collar C (F) to maintain the motion is
Explanation of Solution
Given information:
The weight of the bob
The weight of the collar
The length of the simple pendulum (l) is
The amplitude of the collar
The magnitude of static deflection
The frequency
The acceleration due to gravity (g) is
Calculation:
Consider the collar as a free body and show the free body diagram equation as in Figure (2).
Refer Figure (2), F is the force applied to the collar to maintain the motion is F and
The expression for the force balance equation from Figure 2 as follows:
Substitute mg for T and
Substitute
The expression for the relation for
Differentiate the relation for
Substitute
Substitute
Therefore, the force that must be applied to collar C (F) to maintain the motion is
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Chapter 19 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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