- #1

jonjacson

- 453

- 38

## Homework Statement

To find the period of oscillation of the pendulum and the equation of motion.

## Homework Equations

Conservation of energy.

Potential energy in a constant field = mgh.

Kinetic energy in polar coordinates with r constant = (1/2) m r

^{2}(dΘ

^{2}/dt

^{2})

## The Attempt at a Solution

I won't use the Newton laws because that means a second order differential equation. Instead I use the conservation of energy so I start from a first order equation.

1.- I set the origin of coordinates at the anchor point of the pendulum.

2.- I will use polar coordinates, the angle starts counting from the vertical counterclockwise.

3.- The energy is conserved so at the point 1 (at the bottom when the pendulum is faster) we have:

L is the length of the pendulum

KE =(1/2) mL

^{2}dΘ

^{2}/dt

^{2}

Potential energy = -mgL ; I set the potential energy as zero at the origin of coordinates, the anchor point of the pendulum.

At the point 2 when the speed is zero I have:

KE = 0

PE = -mg (lcosΘ)

Now I equate the sum of the KE and PE at both points:

(1/2)mL

^{2}dΘ

^{2}/dt

^{2}-mgL = 0 -mg (LcosΘ)

(1/2) dΘ

^{2}/dt

^{2}= g(1 - cos Θ ) /L

If Θ is small cos Θ can be aproximated by 1 - Θ

^{2}/2, so:

(1/2) dΘ

^{2}/dt

^{2}= g ( 1- (1-Θ

^{2}/2))/L

Finally:

dΘ/dt = √(g/L) Θ

^{2}

dΘ/√(g/L) Θ

^{2}= dt

Since Θ can get outside de square the integral is :

Log Θ = √L/g t + constant

And the basic expression is an exponential:

So Θ= exp

^{t}

And this is not the expected periodic movement and that does not make any sense. Does anybody know where is the mistake?

THanks