To approximate the maximum percentage error in the calculated value of the period $\tau$ of a pendulum using differentials, we can follow these steps:

Step 1: Express $\tau$ in terms of $r$ and $g$

Given:

$\tau = 2\pi \sqrt{\frac{r}{g}}$

Step 2: Compute the differential of $\tau$

The differential $d\tau$ approximates the change in $\tau$ due to small changes in $r$ and $g$. To find $d\tau$, we differentiate $\tau$ with respect to $r$ and $g$:

$d\tau = \frac{\partial \tau}{\partial r} dr + \frac{\partial \tau}{\partial g} dg$

First, find the partial derivatives:

$\frac{\partial \tau}{\partial r} = 2\pi \cdot \frac{1}{2} \cdot \left(\frac{r}{g}\right)^{-1/2} \cdot \frac{1}{g} = \frac{\pi}{\sqrt{rg}}$

$\frac{\partial \tau}{\partial g} = 2\pi \cdot \frac{1}{2} \cdot \left(\frac{r}{g}\right)^{-1/2} \cdot \left(-\frac{r}{g^2}\right) = -\frac{\pi r}{\sqrt{rg} \cdot g^2} = -\frac{\pi \sqrt{r}}{2 g^{3/2}}$

Thus:

$d\tau = \frac{\pi}{\sqrt{rg}} dr - \frac{\pi \sqrt{r}}{2 g^{3/2}} dg$

Step 3: Relative Error Calculation

The relative error in $\tau$ is given by:

$\frac{d\tau}{\tau} = \frac{1}{\tau} \left( \frac{\partial \tau}{\partial r} dr + \frac{\partial \tau}{\partial g} dg \right)$

Substitute $\tau = 2\pi \sqrt{\frac{r}{g}}$ into the above expression:

$\frac{d\tau}{\tau} = \frac{1}{2\pi \sqrt{\frac{r}{g}}} \left( \frac{\pi}{\sqrt{rg}} dr - \frac{\pi \sqrt{r}}{2 g^{3/2}} dg \right)$

Simplifying:

$\frac{d\tau}{\tau} = \frac{1}{2} \cdot \frac{dr}{r} - \frac{1}{2} \cdot \frac{dg}{g}$

Step 4: Maximum Percentage Error

Given that the maximum percentage errors in $r$ and $g$ are 0.5% and 0.1% respectively, we can substitute these values:

$\frac{dr}{r} = 0.5\% = 0.005$ $\frac{dg}{g} = 0.1\% = 0.001$

Substitute these into the relative error formula:

$\frac{d\tau}{\tau} = \frac{1}{2} \cdot 0.005 - \frac{1}{2} \cdot 0.001 = \frac{0.005}{2} - \frac{0.001}{2} = 0.002$

Step 5: Express the Result as a Percentage

The maximum percentage error in $\tau$ is:

$\frac{d\tau}{\tau} \times 100\% = 0.002 \times 100\% = 0.2\%$

Conclusion

The maximum percentage error in the calculated value of the period $\tau$ is approximately 0.2%.

Would you like further details or have any questions? Here are some related questions:

1. How would the percentage error change if the error in $g$ were 0.5% instead of 0.1%?
2. What is the physical interpretation of the period $\tau$ in relation to the length $r$ and gravity $g$?
3. How does increasing the length $r$ of the pendulum affect the period $\tau$?
4. Can you derive the formula for the period $\tau$ using basic principles of physics?
5. How would the percentage error in $\tau$ be affected if $r$ and $g$ were measured with higher precision?

Tip: When using differentials to estimate errors, always ensure that the approximation is valid for small errors, as larger errors may require more sophisticated methods.