We are asked to approximate the maximum percentage error in the period $T$ of a simple pendulum using a linear approximation. The period $T$ is given by:

$T = 2\pi \sqrt{\frac{L}{g}}$

where:

• $L$ is the length of the pendulum,
• $g$ is the acceleration due to gravity.

The errors in $L$ and $g$ are at most 0.2% and 0.05%, respectively. We will use a linear approximation (differential method) to estimate the resulting error in $T$.

Step 1: Linear Approximation (Differential)

To use linear approximation, we first compute the total differential of $T$. Taking the partial derivatives of $T$ with respect to $L$ and $g$, we have:

$\frac{\partial T}{\partial L} = \frac{2\pi}{2} \cdot \frac{1}{\sqrt{gL}} = \frac{\pi}{\sqrt{gL}}$ $\frac{\partial T}{\partial g} = -\frac{2\pi}{2} \cdot \frac{L^{1/2}}{g^{3/2}} = -\frac{\pi L^{1/2}}{g^{3/2}}$

Now, the total differential $dT$ is:

$dT = \frac{\partial T}{\partial L} dL + \frac{\partial T}{\partial g} dg$

Substituting the partial derivatives:

$dT = \frac{\pi}{\sqrt{gL}} dL - \frac{\pi L^{1/2}}{g^{3/2}} dg$

Step 2: Relative Errors

We are given the percentage errors in $L$ and $g$. The relative errors are:

\frac{dL}{L} = 0.002 \quad \text{(0.2% error in L)}, \quad \frac{dg}{g} = 0.0005 \quad \text{(0.05% error in g)}.

Now, taking the relative error for $T$:

$\frac{dT}{T} = \frac{1}{2} \frac{dL}{L} + \frac{1}{2} \frac{dg}{g}$

Substituting the given values:

$\frac{dT}{T} = \frac{1}{2} (0.002) + \frac{1}{2} (0.0005) = 0.001 + 0.00025 = 0.00125$

Thus, the maximum relative error in $T$ is $0.00125$, or 0.125%.

Final Answer:

The maximum percentage error in the calculated value of $T$ is 0.125%.

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Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How does the period of a pendulum change with length $L$?
2. What is the effect of a higher gravitational constant $g$ on the period $T$?
3. How would you calculate the error if the pendulum’s mass was also considered?
4. Can you derive the formula for a simple pendulum’s period from basic principles?
5. How would the error analysis change if the error in $g$ were 1% instead of 0.05%?

Tip: In physics, percentage errors are always combined using a linear approximation when small errors are present, as it simplifies the error propagation process.