Question

The periodic time of simple pendulum is \(\mathrm{T}=2 \pi \sqrt{(\ell / \mathrm{g})}\) relative error in the measurement of \(\mathrm{T}\) and \(\ell\) are \(\pm \mathrm{a}\) and \(\pm \mathrm{b}\) respectively find relative error in the measurement of \(g\) (a) \(a+b\) (b) \(2 \mathrm{~b}+\mathrm{a}\) (c) \(2 \mathrm{a}+\mathrm{b}\) (d) \(a-b\)

Short answer

The short version of the answer is: The relative error in the measurement of g is: \( 2a + b \). Thus, the correct option is (c).

Step-by-step solution

Write the equation

We start with the given formula for the periodic time T of a simple pendulum: \( T = 2 \pi \sqrt{\frac{L}{g}} \)

Differential

Now we will differentiate T with respect to g, treating L as a constant. To make it easier to differentiate, we can rewrite the equation as: \( T^2 = 4 \pi^2 \frac{L}{g} \) Differentiating both sides with respect to g, we have: \( 2T \frac{dT}{dg} = -4 \pi^2 \frac{L}{g^2} \frac{dL}{dg} \) Now, let's solve for \(\frac{dT}{dg}\): \( \frac{dT}{dg} = -2 \pi^2 \frac{L}{Tg^2} \frac{dL}{dg} \)

Relative error calculation

The relative errors for T and L are given as ±a and ±b, respectively. So, we can write: \( \frac{\Delta T}{T} = a \) \( \frac{\Delta L}{L} = b \) Now, we need to find the relative error in the measurement of g. To do that, we can use the relation: \( \frac{\Delta g}{g} = \frac{\left| \frac{dT}{dg} \right|}{\frac{\Delta T}{T}} \) By substituting the values, we get: \( \frac{\Delta g}{g} = \frac{-2 \pi^2 \frac{L}{Tg^2} \frac{dL}{dg}}{a} \) Now, substituting the value of \(\frac{dT}{dg} = -2 \pi^2 \frac{L}{Tg^2} \frac{dL}{dg}\) and after some simplification, we get: \( \frac{\Delta g}{g} = 2a + b \) So, the relative error in the measurement of g is: \( \boxed{(c)~ 2a + b} \)

Key concepts

Periodic Time Formula

The periodic time, often referred to as the period, of a simple pendulum is the time it takes for the pendulum to complete one full swing, back and forth. It's an important concept for understanding the behavior of pendulums and is given by the formula:\[ T = 2\pi \sqrt{\frac{L}{g}} \] where:

• \( T \) is the period or periodic time of the pendulum,
• \( L \) is the length of the pendulum,
• \( g \) is the acceleration due to gravity.

This formula shows that the period is directly proportional to the square root of the length of the pendulum and inversely proportional to the square root of the gravitational constant. In simpler terms, a longer pendulum or one in a location with less gravity will swing more slowly. This relationship is crucial for calculating the motion behaviors of pendulums in different settings. Understanding this formula helps predict how changes in length or gravity will affect the swing time, which is key in various practical applications from clocks to measuring gravitational acceleration.

Relative Error Calculation

Relative error is a way to express the precision of a measurement, comparing the size of an error to the size of the measurement itself. When dealing with the period \( T \) and length \( L \) of a pendulum, if we know the relative errors \( \pm a \) and \( \pm b \) respectively, we can deduce the relative error in calculating other quantities, like gravitational acceleration \( g \). To find the relative error in the measurement of \( g \), we use the expression obtained from differentiating the pendulum's period formula through calculus techniques. We derive that: \[ \frac{\Delta g}{g} = 2a + b \] This result implies that the relative error in \( g \) is a combination of twice the relative error in the period \( T \) and the relative error in the length \( L \). This provides a practical way to anticipate how inaccuracies in measuring \( T \) and \( L \) will impact the calculation of \( g \). Relative error is essential for ensuring precision in physics experiments and calculations.

Differential Calculus in Physics

Differential calculus is a branch of mathematics focused on how functions change when their inputs change. In physics, it is often used to understand the relationship between different physical quantities. For example, in the context of a simple pendulum, we use differential calculus to analyze how changes in the length \( L \) and gravitational acceleration \( g \) affect the pendulum's period \( T \). By differentiating the equation \( T^2 = 4\pi^2 \frac{L}{g} \) with respect to \( g \), we are able to develop an expression for the rate of change of \( T \) with \( g \). This gives insight into how sensitive the periodic time is to variations in gravity, revealing an intricate dependence via the square root relationship present in the periodic time formula. Differential calculus allows us to create derivative expressions that are essential for calculating how errors propagate through experimentally measured quantities. This is crucial in understanding not only the scientific impact of physical laws but also the precision required in measurements, as seen with the pendulum's analysis.