Question

The pressure on a square plate is measured by measuring the force on the plate and the length of the sides of the plate by using the formula $p = \dfrac{F}{{{l^2}}}$. If the maximum errors in the measurement of force and length are $4\% $and $2\% $ respectively, then the maximum error in the measurement of pressure is:
(A) $1\% $
(B) $2\% $
(C) $8\% $
(D) $10\% $

Answer 1

Hint: Here we are given the formula for pressure on a square plate in the terms of force and length. Both the force and length have some errors in it. It is given that the error in the force is given as $4\% $and the error in the length is given as$2\% $. We have to find the error in the pressure obtained using these two values.

Complete step by step solution:
The pressure on the square plate is given by the expression,
$p = \dfrac{F}{{{l^2}}}$
Where $F$is the force and $l$stands for the length of the plate.
The error in force is given as $4\% $
The percentage error in $F$can be written as, $\dfrac{{\Delta F}}{F} \times 100 = 4\% $
The error in length is given as $2\% $
The percentage error in the length can be written as, $\dfrac{{\Delta l}}{l} \times 100 = 2\% $
The error in pressure can be written,
  $\dfrac{{\Delta p}}{p} \times 100 = \dfrac{{\Delta F}}{F} \times 100 + 2 \times \dfrac{{\Delta l}}{l} \times 100$
Substituting the values, we get
$\dfrac{{\Delta p}}{p} \times 100 = 4 + 2 \times 2 = 8\% $

The answer is: Option (C): $8\% $

Note:
When we use instruments to measure a quantity, the measured value will be different from its true value. The difference between the measured value and the real value is called the error. The errors are broadly classified into two categories that are the systematic errors and random errors. Systematic errors can happen due to the errors of the instrument or due to the error in the procedure or due to personal error. Such kind of errors will propagate in the same direction. The errors that occur irregularly are called random errors. These errors may be caused by random and unpredictable fluctuations in the experimental conditions. Variations in temperature, voltage fluctuations, etc. can cause random errors.