In an experiment, the following observation's were recorded : L = 2.820 m, M = 3.00 kg, l = 0.087 cm, Diameter D = 0.041 cm Taking g = 9.81 m / s² using the formula , Y= 4Mg/πD²l , the maximum permissible error in Y is

To find the maximum permissible error in the Young's modulus ($Y$), we start with the given formula:
$Y=\frac{4Mg}{\pi D^{2}l}$

Here, the acceleration due to gravity ($g$) and the numerical constants (4 and $\pi$) are assumed to be constants without experimental errors. Therefore, the variables that contribute to the experimental error in $Y$ are the mass $M$, the diameter $D$, and the extension $l$.

The relative error in $Y$ is given by the sum of the relative errors of its constituent measured quantities, multiplied by their respective powers:
$\frac{\Delta Y}{Y}=\frac{\Delta M}{M}+2\frac{\Delta D}{D}+\frac{\Delta l}{l}$

To express this as a percentage error, we multiply both sides by 100:
$\frac{\Delta Y}{Y}\times 100=\left(\frac{\Delta M}{M}+2\frac{\Delta D}{D}+\frac{\Delta l}{l}\right)\times 100$

Now, let's identify the values and their least counts (absolute errors, which are determined by the precision of the measurements, represented by the last decimal place):
- Mass, $M=3.00\text{kg}$ ⇒ Least count $\Delta M=0.01\text{kg}$
- Diameter, $D=0.041\text{cm}$ ⇒ Least count $\Delta D=0.001\text{cm}$
- Extension, $l=0.087\text{cm}$ ⇒ Least count $\Delta l=0.001\text{cm}$

Substitute these values into the error formula:
$\frac{\Delta Y}{Y}\times 100=\left(\frac{0.01}{3.00}+2\times \frac{0.001}{0.041}+\frac{0.001}{0.087}\right)\times 100$

Calculate each term separately:
- Relative error in $M$: $\frac{0.01}{3.00}\times 100\approx 0.333\%$
- Relative error in $D$: $2\times \frac{0.001}{0.041}\times 100\approx 2\times 2.439\%=4.878\%$
- Relative error in $l$: $\frac{0.001}{0.087}\times 100\approx 1.149\%$

Summing these percentage errors:
$\text{Percentage Error in}Y\approx 0.333\%+4.878\%+1.149\%=6.36\%$

Rounding to the closest standard option provided, the maximum permissible error in $Y$ is approximately 6.50%.