Question

Suppose a simple pendulum is used to measure the acceleration due to gravity at various points on the Earth. If \(g\) varies by \(0.16 \%\) over the various locations where it is sampled, what is the corresponding variation in the period of the pendulum? Assume that the length of the pendulum does not change from one site to another.

Short answer

Answer: -0.08%

Step-by-step solution

Formula for the period of a simple pendulum

The period of a simple pendulum is given by the formula: $$ T = 2\pi\sqrt{\frac{l}{g}}$$ Where \(T\) is the period, \(l\) is the length of the pendulum, and \(g\) is the acceleration due to gravity.

Differentiate the period with respect to \(g\)

To find the variation in the period, we differentiate the period (\(T\)) with respect to \(g\). We obtain the following: $$ \frac{dT}{d_g} = -\frac{\pi}{g}\sqrt{\frac{l}{g}}$$

Calculate the relative variation in the period

The relative variation in the period can be expressed as: $$ \frac{dT}{T} = \frac{\frac{dT}{d_g}}{T} $$ Substitute the values of \(\frac{dT}{d_g}\) and \(T\) in the above equation using the formulas derived in Step 2 and Step 1: $$ \frac{dT}{T} = \frac{-\frac{\pi}{g}\sqrt{\frac{l}{g}}}{2\pi\sqrt{\frac{l}{g}}} = -\frac{1}{2} \frac{d_g}{g} $$

Solve for the variation in the period

We are given that the variation in \(g\) is \(0.16 \%\), which means \(\frac{d_g}{g} = 0.0016\). Now solve for the variation in the period using the equation derived in Step 3: $$\frac{dT}{T} = -\frac{1}{2} \cdot 0.0016 = -0.0008$$ So, the variation in the period is \(-0.0008\) or \(-0.08 \%\). This means that the period of the pendulum decreases by \(0.08 \%\) as the acceleration due to gravity varies by \(0.16 \%\).

Key concepts

Acceleration due to gravity

Acceleration due to gravity, commonly represented by the symbol \( g \), is the acceleration experienced by an object when the only force acting on it is Earth's gravitational force. Standard gravity, or \( g_0 \), is the average acceleration due to gravity at Earth's surface and is approximately 9.81 meters per second squared (\( m/s^2 \)). However, \( g \) is not constant and can vary slightly depending on altitude, latitude, and local geological structures. The variations in \( g \) are important as they affect the motion of pendulums, the trajectory of projectiles, and calibration of sensitive instruments.

Period of a simple pendulum

The period of a simple pendulum, denoted by \( T \), is the time it takes for the pendulum to make one complete oscillation. It is given by the formula \( T = 2\pi\sqrt{\frac{l}{g}} \), where \( l \) represents the length of the pendulum, and \( g \) is the acceleration due to gravity at the pendulum's location. The formula illustrates that the period is independent of the mass of the bob and is directly influenced by the pendulum's length and the value of \( g \). An important insight, which is often counterintuitive, is that the period of the pendulum is independent of the amplitude of the swing, as long as the angles are small and the motion approximates simple harmonic motion.

Differential calculus in physics

Differential calculus is a mathematical tool widely used in physics to analyze the rate at which quantities change. The differentiation process helps us understand how small changes in one variable can impact another variable. For instance, by differentiating the period of a simple pendulum with respect to \( g \), we can determine how sensitive the pendulum’s period is to changes in the local acceleration due to gravity. This concept is crucial in optimizing systems and understanding the behavior of physical phenomena under varying conditions. Differential calculus forms the foundation for many key theories in physics, including mechanics and electromagnetism, where rate changes are substantively linked with fundamental laws of nature.

Measuring gravitational acceleration

Measuring gravitational acceleration is essential in physics as it allows us to understand Earth's gravity field and calibrate instruments accordingly. One of the methods used for this purpose is timing the period of a simple pendulum. Because the period of a pendulum is dependent on the local value of \( g \), precise measurements of the period can yield the acceleration due to gravity. In addition, by observing how the period changes from one location to another, we can infer the variations in local gravitational acceleration. This knowledge is beneficial for geologists, civil engineers, and other scientists who work with phenomena influenced by gravity. The technique provides an elegant example of how classical mechanics can still be utilized for practical, modern-day applications.