Understanding Error in Measurement for Simple Pendulum Experiment

[AStaunton]

Hi there

when doing an experiment with my university associated with my university related to a simple pendulum, I was not confident when calculating the error there was in my measurement, the relevant equation is:

$$T=2\pi\sqrt{\frac{L}{g}}$$

where T=period L=length of pendulum and g=accel due to grav

my problem is with deciding the error in the period, I measure the length L which was 50cm+.5cm and took g as 9.8+.1 as this is a product (L/g) the general idea is we add the percentage error and of course if we are adding two quantities for example (distance1 + distance2) we add the absolute error.
But what I am not sure of is; does taking the square root of the expression (L/g) have any bearing on the error...also does multiplying by 2pi change the error, I think the multiplying by 2pi might for the following reason:
when you want the width of a piece of paper, it is better to measure 50 pieces of paper so we can then divide by 50 and reduce the error by a factor of 50...so I think it is possible that multiplying by 2pi might increase error by a factor of 2pi..

another example is if I measure the angle theta to accuracy of +1degree and then take the sine of theta, does taking the sine have an effect on the error?

I would be grateful if someone could answer these questions and also suggest rules of thumb when calculating error in more complicated expressions, as I am a third year university physics student now and am too embarassed to ask any of my teachers for advice on something I should have learned years ago!

 

[HallsofIvy]

The simplest way to handle that is to use the differential. From $T= 2\pi L^{1/2}g^{-1/2}$, we have $$dT=\pi g^{-1/2}L^{-1/2}dL- \pi L^{1/2}g^{-3/2}dg$$. You are saying that L= 50 cm.= 0.5 m, dL= 0.5 cm= .005 m, g= 9.81 m/s^2, and dg= .1 m/s^2.

 

[Andy Resnick]

I would be grateful if someone could answer these questions and also suggest rules of thumb when calculating error in more complicated expressions, as I am a third year university physics student now and am too embarassed to ask any of my teachers for advice on something I should have learned years ago!

As HallsofIvy mentioned, the way to determine error propagation is to take the differential. When the different measurements are independent, the errors add in quadrature.

This is the essential book to read:

https://www.amazon.com/dp/093570275X/?tag=pfamazon01-20

 

[Vanadium 50]

The Taylor book is excellent. One case where you can judge a book by its cover.