Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

The period \(T\) of a pendulum of length \(L\) is given by \(T=2 \pi \sqrt{L / g}\), where \(g\) is the acceleration of gravity. Show that \(d T / T=\frac{1}{2}[d L / L-d g / g]\), and use this result to estimate the maximum percentage error in \(T\) due to an error of \(0.5 \%\) in measuring \(L\) and \(0.3 \%\) in measuring \(g\).

Step-by-Step Solution

Answer

The maximum percentage error in the period \(T\) is 0.1\%.

Step 1: Formula Differentiation

First, we need to differentiate the formula for the period of a pendulum with respect to both variables, \(L\) and \(g\). The formula is given as:\[ T = 2 \pi \sqrt{\frac{L}{g}} \]Taking natural logarithms on both sides:\[ \ln T = \ln (2 \pi) + \frac{1}{2} \ln L - \frac{1}{2} \ln g \]Differentiating both sides with respect to \(L\) and \(g\), we get:\[ \frac{dT}{T} = \frac{1}{2} \frac{dL}{L} - \frac{1}{2} \frac{dg}{g} \]

Step 2: Understanding the Formula

The derived formula \(\frac{dT}{T} = \frac{1}{2} \left( \frac{dL}{L} - \frac{dg}{g} \right)\) shows how the relative change in period \(T\) depends linearly on the relative changes in length \(L\) and gravity \(g\). The formula implies that an increase in \(L\) leads to an increase in \(T\), while an increase in \(g\) leads to a decrease in \(T\).

Step 3: Calculating Percentage Error

To find the maximum percentage error in \(T\), given percentage errors in \(L\) and \(g\), we substitute \(\frac{dL}{L} = 0.5\% = 0.005\) and \(\frac{dg}{g} = 0.3\% = 0.003\) into the derived formula:\[ \frac{dT}{T} = \frac{1}{2} (0.005 - 0.003) \]Simplifying this, we find:\[ \frac{dT}{T} = \frac{1}{2} (0.002) = 0.001 \]Converting this back into percentage gives us:\[ 0.1\% \]

Step 4: Interpret the Result

This result reveals that the maximum percentage error in the period \(T\) due to the given measurement errors in \(L\) (0.5\%) and \(g\) (0.3\%) is 0.1\%. This means the period \(T\) could be off by 0.1\% from its true value due to these measurement uncertainties.

Key Concepts

Differentiation

In the realm of mathematics, differentiation is a powerful tool we can use to understand how changes in one quantity affect changes in another. It helps express how a function's output changes as the input changes. When applied to real-world situations, it can help in estimating how errors in measurements affect the results.
For instance, consider a pendulum whose period is determined by its length and the gravitational force. The formula given is:

• \(T = 2\pi \sqrt{\frac{L}{g}}\)

This is a functional relationship between the period \(T\), length \(L\), and gravitational constant \(g\). To find how errors in \(L\) and \(g\) affect \(T\), we use differentiation.

• Taking the natural logarithm of both sides yields \(\ln T = \ln (2\pi) + \frac{1}{2} \ln L - \frac{1}{2} \ln g\).
• By differentiating with respect to \(L\) and \(g\), we capture the essence of how \(T\) changes in relation to these variables. This results in the formula: \(\frac{dT}{T} = \frac{1}{2} \frac{dL}{L} - \frac{1}{2} \frac{dg}{g}\).

This expression allows us to compute the relative change, \(\frac{dT}{T}\), in the pendulum's period, an essential insight when precise measurements are necessary.

Pendulum Period

A pendulum's period, which is the time it takes for one complete swing, is fascinating due to its dependence on certain physical parameters. Specifically, the period \(T\) is calculated using the formula:

• \(T = 2\pi \sqrt{\frac{L}{g}}\)

In this formula,
\(L\) represents the length of the pendulum, and \(g\) is the acceleration due to gravity, approximately \(9.81 \text{m/s}^2\) on Earth. A few important points to understand here:

• The formula shows that the period \(T\) increases with the pendulum's length \(L\).
• An increase in gravity \(g\) will result in a shorter period, meaning the pendulum swings faster.

This relationship is critical, especially in scientific instruments or clocks using pendulums, where precise timing is crucial. Understanding each component of this equation helps in better designing and interpreting experiments and devices relying on pendular motion.

Measurement Accuracy

Measurement accuracy is key in science and engineering, especially when using formulas derived from physical laws to predict outcomes from measured inputs. In our exercise, we're tackling this with the pendulum period formula:\(T = 2\pi \sqrt{\frac{L}{g}}\).

• If there's a ±0.5% error in measuring the length \(L\), that's \(\frac{dL}{L} = 0.005\).
• If the gravitational measurement \(g\) has a ±0.3% error, it's \(\frac{dg}{g} = 0.003\).

These measurement inaccuracies directly impact the period \(T\). Using the formula derived from differentiation, we estimate the maximum possible error in \(T\) due to these inaccuracies:

• \(\frac{dT}{T} = \frac{1}{2} (0.005 - 0.003) = 0.001\)
• Converting to a percentage error, \(0.001 \times 100 = 0.1\%\).

Thus, the period could be off by 0.1% due to measurement errors. Understanding this effect helps in ensuring that the instruments or setups are calibrated accurately to minimize errors, enhancing reliability in observations and experiments.