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

In order to calculate the percentage error, we need to follow the formula of the error in question which is percent error= [experimental value – theoretical value]/theoretical value×100%
As the formula of pressure is given $P = \dfrac{F}{{{l^2}}}$ then we can write it in the form of percentage error i.e. $\dfrac{{\Delta P}}{P} \times 100 = \dfrac{{\Delta F}}{F} \times 100 + \dfrac{{2\Delta L}}{L} \times 100$ , on substituting the values we will get the desired result.

Complete Step by step solution
Here, it is given that
Side of square plate = L
Percentage error in $\dfrac{{\Delta L}}{L} = 2\%$
Percentage error in force is $\dfrac{{\Delta F}}{F} = 4\%$
we have to find the maximum error in the measurement of pressure.
For this, we know that the relation between force, pressure and length of square plate is given as
$P = \dfrac{F}{{{L^2}}}$ ……………………. (1)
Where, P is the pressure
F is force and L is the side of the square plate.
As we also know that the formula of error is percent error= [experimental value – theoretical value]/theoretical value×100%, therefore the formula of error for the equation (1) can be written as
$\dfrac{{\Delta P}}{P} \times 100 = \dfrac{{\Delta F}}{F} \times 100 + \dfrac{{2\Delta L}}{L} \times 100$
Now, substitute the given values in above equation, we get
$\Rightarrow \dfrac{{\Delta P}}{P} = \left[ {4 + 2\left( 2 \right)} \right]$
$\Rightarrow \dfrac{{\Delta P}}{P} = 8\%$
Hence, the maximum error in the measurement of pressure is 8%
Thus, option C is correct.

Note:
The purpose of percent error calculation is to gauge how close a measured value is to true value. In some fields the percent error is always expressed as a positive number. In others, it is correct to have the positive or negative value. The sign is kept to determine whether the recorded value consistently falls above or below the expected value.
While solving this question, we need to be more careful with formula for finding the percentage error as it is used in a modified way. This question can also be solved by different ways as per the convenience.