Question

In the experiment of simple pendulum error in length of pendulum \((\ell)\) is \(5 \%\) and that of \(g\) is \(3 \%\) then find percentage error in measurement of periodic time for pendulum (a) \(4.2 \%\) (b) \(1.2 \%\) (c) \(2 \%\) (d) \(4 \%\)

Short answer

The percentage error in the measurement of periodic time for the pendulum is \(4 \%\). The answer is (d) \(4 \%\).

Step-by-step solution

Use the formula for the period of a simple pendulum

The formula for the period \((T)\) of a simple pendulum is given by: \[ T = 2\pi \sqrt{\frac{\ell}{g}} \]

Compute the relative error in the period

To find the relative error in the periodic time, we need to compute the derivative of the period with respect to both \(\ell\) and \(g\). First, let's find the derivative of the period with respect to \(\ell\): \[ \frac{dT}{d\ell} = \frac{2\pi}{2 \sqrt{\ell g}} \] Now, we'll find the derivative of the period with respect to \(g\): \[ \frac{dT}{dg} = -\frac{\pi \sqrt{\ell}}{g^{\frac{3}{2}}} \]

Calculate the percentage error in the period

To find the percentage error in the period \((T)\), we will apply the formula for relative errors. If \(x\) and \(y\) are two variables, the relative error in \(z = f(x, y)\) is given by: \[ \frac{\Delta z}{z} = \left|\frac{\partial f}{\partial x}\frac{\Delta x}{x}\right| + \left|\frac{\partial f}{\partial y}\frac{\Delta y}{y}\right| \] Applying this formula to the period \((T)\), we get: \[ \frac{\Delta T}{T} = \left|\frac{dT}{d\ell}\frac{\Delta \ell}{\ell}\right| + \left|\frac{dT}{dg}\frac{\Delta g}{g}\right| \] Substituting the known values of percentage errors in \(\ell\) and \(g\) and the derivatives of the period with respect to \(\ell\) and \(g\): \[ \frac{\Delta T}{T} = \left|\frac{2\pi}{2 \sqrt{\ell g}}\frac{0.05 \ell}{\ell}\right| + \left|-\frac{\pi \sqrt{\ell}}{g^{\frac{3}{2}}}\frac{0.03 g}{g}\right| \] Simplifying the equation further: \[ \frac{\Delta T}{T} = \left|\frac{\pi}{\sqrt{\ell g}}(0.05)\right| + \left|\frac{\pi \sqrt{\ell}}{g^{\frac{3}{2}}}(0.03)\right| \] Notice that \(T = 2\pi \sqrt{\frac{\ell}{g}}\). Therefore, we can write the above equation as: \[ \frac{\Delta T}{T} = \frac{0.05}{2} + \frac{0.03}{2} \] Now, let's find \(\frac{\Delta T}{T}\): \[ \frac{\Delta T}{T} = 0.025 + 0.015 = 0.04 \] So, the percentage error in the measurement of periodic time for pendulum is \(4 \%\). The answer is (d) \(4 \%\).

Key concepts

Period of a Pendulum

The period of a simple pendulum is a fundamental concept in physics. It represents the time taken for the pendulum to complete one full swing back and forth. The formula to calculate the period \(T\) of a simple pendulum is \( T = 2\pi \sqrt{\frac{\ell}{g}} \), where \(\ell\) is the length of the pendulum, and \(g\) is the acceleration due to gravity.
The formula shows that the period depends on two main factors: the length of the pendulum and the gravitational force. This means that if you change the length of the pendulum or if you're in a location with a different gravitational pull, the period will change.
Interestingly, the mass of the pendulum bob doesn't affect the period, which makes this an excellent example of simple harmonic motion. Understanding how the period is calculated is crucial for analyzing the motion of pendulums in clocks, seismographs, and engineering applications.

Percentage Error

When measuring physical quantities, it's essential to consider the accuracy of your measurements. The percentage error helps in understanding how precise these measurements are. In the context of pendulums, if there is an error in measuring the length \(\ell\) or the acceleration due to gravity \(g\), it will affect the calculated period \(T\).
To find the percentage error in the period, we use the formula for relative errors. In our problem, we had a 5% error in the length \(\ell\) and a 3% error in \(g\). By applying the rules of percentage error propagation, we can see how these individual errors affect the overall measurement. This is crucial in experiments where precision is key.

• It's calculated by the formula \( \frac{\Delta T}{T} = \frac{\Delta \ell}{\ell} + \frac{\Delta g}{g} \), where \(\Delta T\) is the error in period.
• For pendulums and many other systems, **small errors** in measurement can lead to significant deviations in calculated results.

Overall, understanding and calculating percentage errors help in evaluating the reliability of an experiment's results.

Derivatives in Physics

Derivatives help us understand how a function changes as its input changes. It's a powerful tool in mathematics and physics, particularly for analyzing dynamic systems.
In the context of a pendulum, we differentiate the period equation with respect to both \(\ell\) and \(g\) to determine how small changes in these variables affect the period \(T\).

• The derivative of the period with respect to \(\ell\) is \( \frac{dT}{d\ell} = \frac{\pi}{\sqrt{\ell g}} \).
• The derivative with respect to \(g\) is \( \frac{dT}{dg} = -\frac{\pi \sqrt{\ell}}{g^{3/2}} \).

By using these derivatives, we can assess the sensitivity of the period to changes in length and gravitational acceleration.
Utilizing derivatives helps in optimizing designs in engineering and physics, especially in systems that rely on precise timing and motion.

Measurement Error Analysis

In any experimental setup, especially in physics, measurement errors are unavoidable. Measurement error analysis is the study and characterization of the errors introduced in measurements.
Errors can stem from a variety of sources, such as instrument precision limits, observer interpretation, or environmental factors. In the case of the pendulum:

• Errors in measuring the length \(\ell\) could be due to a misread ruler or an unstable pendulum setup.
• Errors in determining \(g\) often arise from using approximate values or measuring in areas with variable gravitational fields.

Thus, assessing these errors is paramount for improving the accuracy of experimental results.
Analyzing measurement errors allows for refinement in method and instrumentation, leading to higher precision in future experiments and a deeper understanding of the physical phenomena being investigated.