Deriving the Pendulum Time Period Formula from "Fundamentals of Physics

In summary, Young and Freedman present a series approximation to the exact solution to the period of a simple pendulum as a function of the initial angle. For angles that do not satisfy the usual small angle approximation, sin(θ/2) = θ. When θ is very small, sin(θ/2) = θ. The result is θ=Asin(sqr(g/l)*t) A is Amplitude. Vibration cycle is 2π*sqr(l/g).
  • #1
Ali Asadullah
99
0
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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  • #2
Can you please post the formula here so we'll have a more clear idea of what you are talking about?
 
  • #3
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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  • #4
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
 
  • #5
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)
 
  • #6
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.
 
  • #7
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
 

What is the Pendulum Time Period Formula?

The Pendulum Time Period Formula is a mathematical equation that calculates the time it takes for a pendulum to complete one full swing. It is derived from the principles of harmonic motion and is expressed as T = 2π√(L/g), where T is the time period, L is the length of the pendulum, and g is the acceleration due to gravity.

What are the fundamental principles used to derive the Pendulum Time Period Formula?

The Pendulum Time Period Formula is derived from two fundamental principles of physics: harmonic motion and the law of conservation of energy. Harmonic motion refers to the repetitive pattern of motion that a swinging pendulum follows. The law of conservation of energy states that energy cannot be created or destroyed, only transferred from one form to another. By applying these principles, the formula for the pendulum time period can be derived.

How does the length of the pendulum affect the time period?

The length of the pendulum directly affects the time period. As the length increases, the time period also increases. This is because a longer pendulum has a greater distance to cover in each swing, resulting in a longer time period. Conversely, a shorter pendulum will have a shorter time period.

Why is the acceleration due to gravity used in the Pendulum Time Period Formula?

The acceleration due to gravity, denoted as g, is used in the Pendulum Time Period Formula because it is responsible for the restoring force that allows the pendulum to swing back and forth. This force is directly proportional to the acceleration due to gravity, meaning that a higher value of g will result in a faster swinging motion and a shorter time period.

Can the Pendulum Time Period Formula be applied to all pendulums?

No, the Pendulum Time Period Formula can only be applied to simple pendulums, which are pendulums that have a mass attached to a weightless string or rod. Other factors, such as air resistance and the mass distribution of the pendulum, can affect the time period and therefore, cannot be accounted for in the formula. Complex pendulums may require different equations to calculate their time period.

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