# The period of a pendulum

1. Sep 24, 2006

### phlegmy

hey guys i think its well know the period of a pendulum is 2*pi*root(l/g)
which works fine except!...
i've written a small program to model how the bob of a pedulum moves, due to the net forces experienced by it, namely the tension in the wire which is the result of the bobs weight and the centripedal force due to is angular velocity. and the results my prog has spat out are interesting

*when the bob is initially disturbed from its zero position by a very small angle the period is indeed matched my 2*pi*root(l/g)

however
when the bob is initally disturbed by a significant angle (anything from 10 to 89.9 degrees) the period returned by the program is larger than
2*pi*root(l/g)

also the larger the inital disturbance the longer the period becomes.

now what i really want to know is have i made a mistake in my programming!
or
is the reason for the period not agreeing with 2*pi*root(l/g), that the derivation of this formula does not include centripedal force as its assuming it to be so small, due to angular velocity being small, due to the initial disturbace being so small?
and if so then why dont they tell you that in school or college!

all replies appreciated

2. Sep 24, 2006

### arildno

The period of the pendulum is independent of the initial angle only for tiny opening angles.

3. Sep 24, 2006

### Mattara

The approximation gives 1% deviation at an angle of 23 degrees.

4. Sep 24, 2006

### arildno

Last edited: Sep 24, 2006
5. Sep 24, 2006

### phlegmy

wow thanks for the quick replies
good to hear that the problem may not be with my programme
as for 1% deviation for a 23 degree starting angle
i see this increase is given by the formula in the link and the ratio of the period given by the small angle formula and this one, is independant of the length of the pendulum
but....
my prog gives a period of +3% for a lenght of .5 meters
of +3% for a length of 2 metres
and of +3% for a length of 20m
all for a starting angle of 23 degrees

so i'm guessing the approximation formula is pretty accurate at +1% for 23 degrees. so where then am i going wrong
i've included the weight and the centripedal force as (and resulting tension) as the only influences on the bob of the pendulum, is there anything i'm leaving out (i know friction, but i'm assuming the approximation formula does not count on this)

6. Sep 24, 2006

### Integral

Staff Emeritus
What method are you using in your numerical scheme?

7. Sep 24, 2006

### phlegmy

the programme works on a loop
each loop advancing time by 5 milliseconds

1>i start with the horizontal displacement of the bob and angular velocity (0)

2>from this and knowing length i find vertical displacement

3>then from the angle of the bob and its angular velocity i get tension in wire

4>from this i get the horizontal force on the bob

5>its subsequent horizontal veloity after 5ms

6>its subsequent horizontal postion after 5ms

7>the new vertical positon from this new horizontal position

8>the new angular velocity from its new position and last position

9>the new tension in the wire from its position and angular velocity

then back to step 3 and reiterate ad infinitum recording the position of the bob and the "time" at each iteration

8. Sep 24, 2006

### Claude Bile

The problem is not with the program.

The formula for the period of a pendulum is actually derived from the equation of motion of a mass rotating around a point in the presence of a restoring force. In deriving this result however, the small angle approximation (i.e. sin(x) = x) is made, making the equation inaccurate for large angles.

To put it another way, the equation describing the motion of the pendulum is linear, however the system is actually a rotational one. We can, however approximate the rotational system as being linear as long as we keep the angle small, much the same way as the Earth is approximately flat for short distances.

To make your model work for larger angles, you need to use the original equation of motion for a mass rotating about a point with some restoring force.

Claude.

9. Sep 24, 2006

### phlegmy

thanks claude!
thats actually how i approached the problem, with the tension in the "string" of the pendulum being a restoring force twards the point of suspension, i didnt linearise it at all, which is why the classic "small angle" pendulum formula was throwing me a bit. anway i'm happy enough with the programme, i've just gotten the point of suspension of the pedulum to accelerate linearly and its returning usefull info on how the bob will react!

i'm hoping to develop a model of a tower crane for my final year project in mechanical engineering, which will keep the load from swaying when brought to rest, by adjusting the trolly on the boom to counter any oscillations. so i was keen to pin down the dynamics of a pendulum!
thanks everyone!

10. Sep 24, 2006

### masudr

A suggestion might be to formulate the problem in terms of the angular displacement from the vertical, and work with transverse force:

i.e. the real eqn. of motion is

$$\ddot{\theta} = -\mbox{\frac{g}{l}}\sin{\theta},$$

but at least for small angles, you can take that to be

$$\ddot{\theta} = -\mbox{\frac{g}{l}}\theta,$$

as opposed to the further simplification that:

$$\ddot{x} = -\mbox{\frac{g}{l}}x,$$

where x is a Cartesian coordinate.

Last edited: Sep 24, 2006
11. Sep 25, 2006

### Integral

Staff Emeritus
I would also suggest looking into some of the standard numerical methods, The forth order Runga-Kutta is very good. You would loose the physical significance which your calculations have but gain much in the way of understanding the errors.