Genesis of the pendulum formula

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SUMMARY

The discussion focuses on the derivation of the pendulum period formula, specifically addressing the forces acting on a pendulum bob at a small angle of 3°. The tension in the rope is calculated as g/cos(3°), with horizontal tension equating to 0.5 N. The conversation emphasizes the need for integration to determine the time taken for the bob to reach the vertical position, given a length of 0.98 m and a final velocity of 16 cm/s. Key concepts include the small angle approximation and the identification of the pendulum as a simple harmonic oscillator.

PREREQUISITES
  • Understanding of basic physics concepts such as tension and gravitational force
  • Familiarity with the small angle approximation in trigonometry
  • Knowledge of simple harmonic motion and oscillators
  • Basic calculus skills for integration
NEXT STEPS
  • Study the derivation of the pendulum period formula, focusing on √(l/g)
  • Learn about the small angle approximation and its applications in physics
  • Explore integration techniques for non-constant acceleration scenarios
  • Investigate the principles of simple harmonic motion and its mathematical representation
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Physics students, educators, and anyone interested in understanding the dynamics of pendulum motion and the mathematical principles behind it.

alba
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Can you give me a link where I can find a simple explanation of the formula of the period of a pendulum?

As far as I know, if the angle is 3°, the tension on the rope is g/ cos 3 and the horizontal T = 9,8 * tan 3 (0.0524) = 0.5 N. Is this the only force to consider?

Can you tell me how to proceed to find the time the bob reaches the vertical position? supposing l = .98 m , I know (from PE) that final v there is 16 cm/s and space traveled is 5.13 cm, but I cannot use the formula: t^2 = 2 s/a: since it is not an incline, a is not constant,

is integration the correct procedure?
of what function?
what is the formula for an arc?

Thanks, as I said, a link will do, how to find the exact value? how do we end up with √l/g?.
 
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You should be able to do this one for yourself with a few hints.

Don't forget that the tension in the rope has a direction. So it tends to pull the mass back towards the vertical orientation of the rope. So you need to work out what that force is as a function of angle. Then you need to take the approximation that the angle is very small, and use the small angle formula for trig functions. Then you should be able to recognize a simple harmonic oscillator.

https://en.wikipedia.org/wiki/Small-angle_approximation
https://en.wikipedia.org/wiki/Harmonic_oscillator
 

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