How Can We Prove the Time Period Formula of a Simple Pendulum?

[Michaelcarson11]

Homework Statement

As part of my coursework, I need to be able to prove the formula for the time period of a simple pendulum. I know what the formula is but I really don't know how to prove it. I would appreciate any help.

Homework Equations

T = 2*PI*Root(l/g)

The Attempt at a Solution

I have not got very far at all with the proof. I worked out that the force back towards the rest position is -mgsinx and we were told that we needed to use the approximation sinx= x, but I cannot get any further

 

[HallsofIvy]

It's not clear to me what the problem is. You say are to "prove the formula for the time period of a simple pendulum." But then you give as a "relevant equation" precisely that formula. Do you mean you want to derive that equation? From what hypotheses? What I would do is set up the formula "F= ma" in terms of the forces on the pendulum, then solve that equation and derive the formula from that. From what you say, you have derived the equation and it should have "sine" in it. The difficulty is that nonlinear equation does not have an exact solution and, in fact, the formula you give for the period is not exactly true. Replacing sin(x) with x (Approximately true for small x. Do you remember from Calculus that sin(x)/x goes to 1 as x goes to 0.) give you a linear equation with linear coefficients. Please show exactly what you have done so far.

 

[ogb p]

than that.


Dear student,

Thank you for reaching out for help with your coursework. I understand the importance of being able to prove formulas and concepts in order to fully understand and apply them. I would be happy to provide some guidance on how to prove the formula for the time period of a simple pendulum.

To begin, let's review the equation for the time period of a simple pendulum: 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. This formula was first derived by Galileo in the 16th century and has been proven through various experiments and mathematical analysis.

One way to prove this formula is by using the principles of Newton's laws of motion. As you mentioned, the force pulling the pendulum back towards its rest position is -mgsinx, where m is the mass of the pendulum and x is the angle of displacement from the rest position. Using Newton's second law, F = ma, we can set the force equal to the mass times the acceleration, and rearrange the equation to solve for acceleration: a = -gsinx.

Next, we can use the small angle approximation, sinx ≈ x, for small values of x. This approximation allows us to simplify the equation to a = -gx.

Now, we can use the equation for acceleration, a = -gx, and the equation for the period of a pendulum, T = 2π√(l/g), to solve for the time period: T = 2π√(l/(-gx)). Simplifying this equation further, we get T = 2π√(l/g), which is the formula for the time period of a simple pendulum.

In conclusion, by using the principles of Newton's laws of motion and the small angle approximation, we can prove the formula for the time period of a simple pendulum. I hope this explanation helps you understand the proof better. If you have any further questions, please do not hesitate to reach out for assistance.

Best regards,