Pendulum experiment?

Question

would you expect to get a more precise g for a longer or shorter pendulum?

why is it important to keep the angle of the string small?

Answer 1

Longer, definately. This way your reaction time will be very small compared to the period of the pendulum. This will make your data more accurate because your reaction time will make only a minor difference and your answer will have less error.

Answer 2

LONGER! Yes, longer every time, for many reasons in addition to reducing time-measuring errors due to your personal reaction time:

1. You can measure its LENGTH L more accurately.

2. The period is longer (P proportional to sqrt (L)), and therefore you can measure the TIME of the swings more accurately. Also:

3. If you've got a mass of a given size on it, you can make any error in determining where its centre is, or any correction that needs to be applied BECAUSE of the finite physical size of the mass, that much relatively smaller. (Remember that the classical formula gives the period for an idealised point mass swinging on a massless string!)

4. You can more easily use and measure a SMALLER angle amplitude with a LONG string than you can with a short one!

It's important to keep the angular amplitude of the swing small because, once again, the classical formula is only valid in the small angle limit. In the differential equation REALLY governing the motion:

d^2 theta / dt^2 + (g/L) sin theta = 0, the approximation

sin theta ≈ theta

is made. That results in a simple harmonic motion (SHM) differential equation,

d^2 theta / dt^2 + (g/L) theta = 0

with Period P given by

P = 2 π (g/L)^(1/2), independently of amplitude.

The approximation of sin theta by theta has a relative error of order [(theta)^2]/6, where 'theta' is in radians. So, the smaller you make theta, the more consistent your results will be.

Live long and prosper.

P.S. Historical note: Recall that Galileo's interest in the pendulum --- which began his fascination with the physics of motion --- was first sparked by watching the swinging of a VERY LONG candelabra in the cathedral where he was a choirboy!