Concept explainers
Bars AB and BC, each with a length l and of negligible weight, are attached to two springs, each of constant k. The springs are undeformed, and the system is in equilibrium when θ1 = θ2 = 0.
Determine the range of values of P for which the equilibrium position is stable.
Fig. P10.97
Find the range of values of P for which the equilibrium of the system is stable.
Answer to Problem 10.97P
The range of values of P for which the equilibrium position is stable is
Explanation of Solution
Given information:
The system is in equilibrium when
Calculation:
Show the free-body diagram of the arrangement as in Figure 1.
Find the horizontal distance
Find the horizontal distance
Find the vertical distance
When the values are small,
Find the potential energy (V) using the relation.
Here, the magnitude of the force applied at C is P and the spring constant is k.
Substitute
Substitute
Differentiate the Equation (1) with respect to
Differentiate the Equation (2).
Differentiate the equation (2) with
Differentiate the Equation (1) with respect to
Differentiate the Equation;
Condition 1:
When the equilibrium is stable,
Substitute 0 for
Substitute 0 for
The condition is satisfied. The equilibrium is stable.
Condition 2:
Check the condition,
Substitute
Solve the equation using the mathematical equation.
Condition 3;
Check the condition;
Condition 4:
Refer to all the conditions,
The minimum value of P is 0.
The maximum value of P is
Therefore, the range of values of P for which the equilibrium position is stable is
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Chapter 10 Solutions
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