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

Click For Summary
SUMMARY

The discussion focuses on proving the time period formula for a simple pendulum, expressed as T = 2π√(l/g). Participants clarify that the goal is to derive this formula using the force equation F = ma and the approximation sin(x) ≈ x for small angles. It is established that the formula is an approximation, valid under the assumption of small angular displacements. The nonlinear nature of the pendulum's motion complicates the proof, as the exact solution is not achievable.

PREREQUISITES
  • Understanding of basic physics concepts, particularly forces and motion.
  • Familiarity with the formula T = 2π√(l/g) for simple pendulums.
  • Knowledge of calculus, specifically the behavior of the sine function near zero.
  • Ability to manipulate and solve differential equations.
NEXT STEPS
  • Study the derivation of the simple pendulum formula using F = ma.
  • Learn about the small-angle approximation and its implications in physics.
  • Explore the concept of nonlinear equations and their solutions in mechanics.
  • Investigate the role of calculus in deriving physical formulas.
USEFUL FOR

Students of physics, educators teaching mechanics, and anyone interested in the mathematical derivation of physical laws related to pendulum motion.

Michaelcarson11
Messages
11
Reaction score
0

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
 
Physics news on Phys.org
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.
 

Thread 'Magnitude of buoyant force in fluids of different densities'

The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?
View full post »

Similar threads

The period of a simple pendulum
  • · Replies 20 ·
Replies
20
Views
2K
Oscillatory motion problem: Bicycle wheel rotation oscillation
  • · Replies 17 ·
Replies
17
Views
1K
Seeking advice on "simple" pendulum problem
  • · Replies 27 ·
Replies
27
Views
2K
JEE solution wrong? (tempco of a clock)
  • · Replies 14 ·
Replies
14
Views
2K
Foucault Pendulum at the North Pole
  • · Replies 9 ·
Replies
9
Views
2K
Derivation of time period for physical pendula without calculus
  • · Replies 3 ·
Replies
3
Views
2K
Understanding T^2 x L Graphing for Simple Pendulum Experiment
  • · Replies 7 ·
Replies
7
Views
5K
Uncertainty of pendulum period and pendulum length
  • · Replies 7 ·
Replies
7
Views
5K
Experiment to determine the time period of a pendulum
  • · Replies 12 ·
Replies
12
Views
2K
How Does Pivot Point Location Affect the Time Period of a Physical Pendulum?
  • · Replies 14 ·
Replies
14
Views
2K