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Solved: Consider the pendulum of Figure 7.4, suspended
Chapter 7, Problem 7.30(choose chapter or problem)
Consider the pendulum of Figure 7.4, suspended inside a railroad car that is being forced to accelerate with a constant acceleration a. (a) Write down the Lagrangian for the system and the equation of motion for the angle 0. Use a trick similar to the one used in Equation (5.11) to write the combination of sin 0 and cos 0 as a multiple of sin( 0 + /3). (b) Find the equilibrium angle 0 at which the pendulum can remain fixed (relative to the car) as the car accelerates. Use the equation of motion to show that this equilbrium is stable. What is the frequency of small oscillations about this equilibrium position? (We shall find a much slicker way to solve this problem in Chapter 9, but the Lagrangian method does give a straightforward route to the answer.)
Questions & Answers
QUESTION:
Consider the pendulum of Figure 7.4, suspended inside a railroad car that is being forced to accelerate with a constant acceleration a. (a) Write down the Lagrangian for the system and the equation of motion for the angle 0. Use a trick similar to the one used in Equation (5.11) to write the combination of sin 0 and cos 0 as a multiple of sin( 0 + /3). (b) Find the equilibrium angle 0 at which the pendulum can remain fixed (relative to the car) as the car accelerates. Use the equation of motion to show that this equilbrium is stable. What is the frequency of small oscillations about this equilibrium position? (We shall find a much slicker way to solve this problem in Chapter 9, but the Lagrangian method does give a straightforward route to the answer.)
ANSWER:Step 1 of 3
The sketch of the railroad car system:
The position of the bob is given by,
Where 
Thus,
