calculate focal lenght of a shperical mirror from the following observation.

Question

object distance u=(50±0.5) cm and images distance v=(20.1±0.2)cm. 

 

to obtain central value is easy. but how to obtain final error in focal lenght.

 

in the book where i found this question there is a solution to this question. they have done it by adding relative errors.

 

but i object. there should be no need to use relative error formula coz the units are same( both u and v are in cm) hence we should be directly able to add the errors. 

 

but when i did so my answers was wrong.

 

so pls help in removing my doubt for this perticular question. 

Answer 1

1/f = 1/u+1/v

It looks like you have this part: f=1/(1/u+1/v)=vu/(v+u)

For the error, it is easier to start from the original formula 1/f=1/u+1/v. Assuming that u and v are independent measurements, then |Δ(1/f)|=Δf/f². Likewise |Δ(1/u)|=|Δu|/u² and |Δ(1/v)|=|Δv|/v². (For this you need to know that d(1/x)/dx=-1/x²). Then you propagate the absolute uncertainty of the sum, which is the square root of the sum of the squares of the uncertainties:

Δf=f²SQRT((Δu/u²)² + (Δv/v²)²)

If u and v are correlated (worst-case error), then it's a simple addition of the two absolute uncertainties of 1/v and 1/u:

Δf=(v/(u+v))²|Δu| + (u/(u+v))²|Δv|

So either way, those are not relative errors in the final expression.

Oleg