A pendulum inside an oscillation railroad car.

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SUMMARY

The discussion centers on the dynamics of a pendulum suspended inside an oscillating railroad car, as outlined in problem 7.11 of Taylor's Classical Mechanics. The Cartesian coordinates of the pendulum bob are expressed as x = lsin(φ) + Acos(ωt) and y = lcos(φ), where φ is the angle of the pendulum. Participants emphasize the importance of defining the orientation of φ and the y-axis to ensure accurate calculations. The solution involves solving for φ in terms of x and y using trigonometric identities.

PREREQUISITES
  • Understanding of pendulum dynamics and oscillatory motion
  • Familiarity with Cartesian coordinate systems
  • Knowledge of trigonometric functions and their applications
  • Proficiency in classical mechanics principles as outlined in Taylor's Classical Mechanics
NEXT STEPS
  • Study the derivation of the equations of motion for pendulums in oscillating frames
  • Learn about the implications of coordinate transformations in dynamic systems
  • Explore the use of trigonometric identities to solve for angles in Cartesian coordinates
  • Review problem-solving techniques in classical mechanics, focusing on oscillatory systems
USEFUL FOR

Students of classical mechanics, physics educators, and anyone interested in the dynamics of oscillating systems and pendulum behavior.

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Homework Statement



A pendulum length l is suspended inside a railroad car. The railroad car is oscillating so that its suspension has position xs = A cos (ωt) and ys= 0. Use the angle \varphi as the generalized coordinate and write down the equations that give the Cartesian coordinates of the bob in terms of \varphi and vice versa.

This is also problem 7.11 in Taylor's Classical Mechanics text.

The Attempt at a Solution



By geometry, thinking of the x and y direction separately...

x= lsin\varphi + Acosωt
y= lcos\varphi

This gives the Cartesian coordinates in terms of \varphi.

But how do I give the \varphi component in terms of x and y? Do I just solve for \varphi?
 
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looks good so far. One thing to watch out for: are you defining phi as "pointing straight up" or "pointing straight down" and is y going to represent "up" or "down" ?
And yep, just solve for phi. Hint: or at least, solve for some trigonometric function of phi.
 

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