Deriving the Pendulum Time Period Formula from "Fundamentals of Physics

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Discussion Overview

The discussion centers around deriving the time period formula for a simple pendulum as presented in "Fundamentals of Physics" by H D Young and Freedman. Participants explore the mathematical foundations of the formula, including its dependence on the initial angle and the use of elliptic integrals, as well as approximations for small angles.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests that the formula involves a series expansion of the sine function, possibly using Maclaurin or Taylor series.
  • Another participant clarifies that the period is expressed as a function of the initial angle and involves solving an elliptic integral, not merely a Taylor series expansion.
  • A specific formula for the time period is provided, involving an integral that represents the complete elliptic integral of the first kind.
  • One participant presents a derivation based on the rigid-body rotation law, leading to a second-order differential equation for small angles, resulting in a simple harmonic motion approximation.
  • Another participant emphasizes that the discussion is about cases where the angle is not small enough for the simple approximation of sin(θ) = θ to hold.
  • Further clarification is provided that the time period formula for the full period is four times the integral presented for a quarter period.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of the small angle approximation and the nature of the derivation, indicating that multiple competing views remain without consensus on the best approach to derive the time period formula.

Contextual Notes

The discussion highlights the complexity of deriving the time period for a simple pendulum, particularly in relation to the initial angle and the mathematical techniques involved, such as elliptic integrals versus Taylor series approximations. There are unresolved aspects regarding the assumptions made in different derivations.

Ali Asadullah
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In the book "Fundamentals of Physics" by H D Young and Freedman,there is formula about the time period of simple pendulum which uses sine function in a complex way which i think is obtained by series expansion of sine function either by Maclurin or Tayler series.
Can anyone please tell me how to derive the formula?
 
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Can you please post the formula here so we'll have a more clear idea of what you are talking about?
 
What Young and Freedman show is the period of a simple pendulum as a function of the initial angle (in other words, for angles that do not satisfy the usual small angle approximation). They present a series approximation to the exact solution, which involves solving an elliptic integral. (It's not a simple Taylor series expansion of a sine function.)

Read this: http://hyperphysics.phy-astr.gsu.edu/HBASE/pendl.html#c1"
 
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The time period is

t = sqrt(b/g) ∫oπ/2 dφ/sqrt[1-k2sin2(φ)]

where k= sin(θ/2), b= radius of pendulum, θ= max angle, and the integral is the complete elliptic integral of the first kind..

Bob S
 
According to Rigid-body rotation Law ，you can get a formula
mglsinθ=-ml^2*(d2θ/dt2)
m is the mass of the ball ,l is the length of the rope and θ is the angle between the rope and vertical.
According to Talyor series ,when θ is very small,sinθ=θ.
Then it turns to (d2θ/dt2)+g/l *θ=0,and it is a Second-order differential equations with constant coefficients. The result is θ=Asin(sqr(g/l)*t) A is Amplitude
It's clearly that Vibration cycle is 2π*sqr(l/g)
 
ftfaaa said:
According to Talyor series ,when θ is very small,sinθ=θ.
We're talking about the case where θ is not small enough for that approximation.
 
Bob S said:
The time period is

t = sqrt(b/g) ∫oπ/2 dφ/sqrt[1-k2sin2(φ)]

where k= sin(θ/2), b= radius of pendulum, θ= max angle, and the integral is the complete elliptic integral of the first kind..
The above answer is actually for a quarter period. The full period is 4 times the above answer:

T = 4·sqrt(b/g) ∫oπ/2 dφ/sqrt[1-k2sin2(φ)]

Bob S
 

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