To deduce the formula for the fractional error in $g$, we'll start with the given formula for the period of oscillation of a simple pendulum:

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

Here:

• $T$ is the period of oscillation,
• $l$ is the length of the pendulum, and
• $g$ is the acceleration due to gravity.

Step 1: Isolate $g$ in the formula

First, we'll isolate $g$ in the formula by squaring both sides to eliminate the square root:

$T^2 = 4\pi^2 \frac{l}{g}$

Next, solve for $g$:

$g = \frac{4\pi^2 l}{T^2}$

Step 2: Apply the concept of fractional error

The fractional error in a quantity is given by the relative error of the quantity. For a function $f(x, y, \dots)$ with variables $x$, $y$, etc., the fractional error is:

$\frac{\Delta f}{f} = \sqrt{\left(\frac{\partial f}{\partial x} \frac{\Delta x}{x}\right)^2 + \left(\frac{\partial f}{\partial y} \frac{\Delta y}{y}\right)^2 + \dots}$

However, since we are dealing with a direct formula, we can differentiate the logarithm of both sides to obtain the fractional errors.

Step 3: Differentiate to find fractional errors

Taking the natural logarithm of the equation $g = \frac{4\pi^2 l}{T^2}$, we get:

$\ln g = \ln(4\pi^2) + \ln l - 2\ln T$

Now, differentiate with respect to $g$, $l$, and $T$ to get the errors:

$\frac{\Delta g}{g} = \frac{\Delta l}{l} + 2\frac{\Delta T}{T}$

Step 4: Interpret the result

The formula above indicates that the fractional error in $g$ depends on the fractional errors in $l$ and $T$. Specifically:

• The fractional error in $l$ directly contributes to the fractional error in $g$.
• The fractional error in $T$ contributes twice as much to the fractional error in $g$ because $T$ appears squared in the original formula.

Thus, the formula for the fractional error in $g$ is:

$\frac{\Delta g}{g} = \frac{\Delta l}{l} + 2\frac{\Delta T}{T}$

Would you like more details or have any questions? Here are some related questions you might explore:

1. How can you minimize the fractional error in a pendulum experiment?
2. What is the physical significance of the fractional error?
3. How would the fractional error change if the length $l$ is measured with higher precision?
4. How does the period $T$ affect the accuracy of the calculated $g$?
5. How does air resistance influence the period of a simple pendulum?
6. What assumptions are made in deriving the period formula for a simple pendulum?
7. How would the formula for $T$ change in a non-uniform gravitational field?
8. What are some practical applications of simple pendulum measurements?

Tip: In experiments, reducing the uncertainty in measurements (e.g., using precise timing methods) is key to minimizing fractional error and obtaining accurate results.