The focal length f of a mirror is given by 1/f=1/u+1/v where u and v represent object and image distances respectively. Find relative error in calculation of focal length if u & v are (100 ± 3)cm and (200 ± 4)cm respectively

Asked by kdimple765 | 9th Jun, 2022, 05:07: PM

Expert Answer:

begin mathsize 14px style 1 over f equals space 1 over u plus 1 over v
end style
begin mathsize 14px style f space equals space fraction numerator u space v over denominator u space plus space v end fraction end style
 
By taking logarithm on both sides of above expression
 
ln(f) = ln( u ) + ln ( v ) - ln(u+v)
 
If we differentiate above expression, we get
 
begin mathsize 14px style fraction numerator d f over denominator f end fraction space equals space fraction numerator d u over denominator u end fraction plus fraction numerator d v over denominator v end fraction space minus fraction numerator d left parenthesis u plus v right parenthesis over denominator u plus v end fraction end style
 
du/u = 3/100  ;   dv/v = 4/200 ;  d(u+v) = du + dv = 3+4 = 7
 
 
begin mathsize 14px style fraction numerator d f over denominator f end fraction space equals space 3 over 100 plus 4 over 200 space minus 7 over 300 space equals space 2 over 75 space equals space 0.027 end style

Answered by Thiyagarajan K | 9th Jun, 2022, 08:40: PM