Question

A Foucault pendulum is designed to demonstrate the effect of the Earth's rotation. A Foucault pendulum displayed in a museum is typically quite long, making the effect easier to see. Consider a Foucault pendulum of length \(15.0 \mathrm{~m}\) with a \(110 .-\mathrm{kg}\) brass bob. It is set to swing with an amplitude of \(3.50^{\circ}\) a) What is the period of the pendulum? b) What is the maximum kinetic energy of the pendulum? c) What is the maximum speed of the pendulum?

Short answer

Question: A Foucault pendulum has a length of 15 meters, a mass of 110 kg, and an amplitude of 3.5°. Calculate the following: a) The period of the pendulum, b) The maximum kinetic energy of the pendulum, c) The maximum speed of the pendulum. Answer: a) The period of the pendulum is approximately 7.73 s, b) The maximum kinetic energy of the pendulum is approximately 153.8 J, c) The maximum speed of the pendulum is approximately 1.75 m/s.

Step-by-step solution

Use the period formula for a simple pendulum

The formula for the period of a simple pendulum is given by: \(T = 2\pi\sqrt{\frac{L}{g}}\) Where \(T\) is the period, \(L\) is the length of the pendulum, and \(g\) is the acceleration due to gravity (approximately \(9.81\,\text{m/s}^2\)).

Plug in the values and calculate the period

Using the given length of \(15.0\,\text{m}\) and the acceleration due to gravity, we can now calculate the period: \(T = 2\pi\sqrt{\frac{15.0}{9.81}}\) After calculating, we find that the period of the pendulum is approximately \(7.73\,\text{s}\). b) Finding the maximum kinetic energy of the pendulum

Calculate the maximum gravitational potential energy

At the highest point of the swing, the pendulum has maximum gravitational potential energy \(U\) given by: \(U = mgh\) where \(m\) is the mass of the pendulum, \(g\) is the acceleration due to gravity, and \(h\) is the height of the pendulum above the equilibrium position.

Find the height of the pendulum

Using the amplitude and the length of the pendulum, we can find the height \(h\) above the equilibrium position: \(h = L - L\cos\text{(amplitude)}\) Substitute the given values: \(h = 15 - 15\cos(3.5^\circ)\) After calculating, we find that the height of the pendulum above the equilibrium position is approximately \(0.143\,\text{m}\).

Compute the maximum gravitational potential energy

Now, we can compute the maximum potential energy using the mass, gravity and height: \(U = (110)(9.81)(0.143)\) After calculating, we find that the maximum gravitational potential energy is approximately \(153.8\,\text{J}\).

Determine the maximum kinetic energy

Because mechanical energy is conserved, the maximum kinetic energy of the pendulum is equal to its maximum gravitational potential energy: \(K_{\text{max}} = U_{\text{max}} = 153.8\,\text{J}\) c) Finding the maximum speed of the pendulum

Use the formula for kinetic energy

We can use the formula for kinetic energy to find the maximum speed \(v\) of the pendulum: \(K = \frac{1}{2}mv^2\)

Rearrange to solve for speed

We can rearrange the kinetic energy formula to solve for \(v\): \(v = \sqrt{\frac{2K}{m}}\)

Plug in the values ​​and calculate the maximum speed

Now, using the maximum kinetic energy and the mass of the pendulum, we can compute the maximum speed: \(v = \sqrt{\frac{2(153.8)}{110}}\) After calculating, we find that the maximum speed of the pendulum is approximately \(1.75\,\text{m/s}\).

Key concepts

Simple Pendulum Period

Understanding the period of a simple pendulum is foundational in the study of oscillatory motion. A simple pendulum consists of a mass, known as the bob, suspended from a pivot point by a string or rod of length \(L\). Its period, \(T\), is the time it takes to complete one full back-and-forth swing. The formula for calculating this period is given by \(T = 2\pi\sqrt{\frac{L}{g}}\), where \(g\) represents the acceleration due to gravity.

This relationship illustrates that the period is independent of the pendulum's mass and its amplitude of swing as long as the amplitude is small. For larger amplitudes, the relationship becomes more complex, but for the Foucault pendulum in the example, with an amplitude of \(3.50^\circ\), the formula approximates the period very well. A longer length \(L\) results in a longer period, meaning that the pendulum swings slower. It is this very characteristic that makes the Foucault pendulum, with its long lengths, an excellent tool for demonstrating the Earth's rotation, as observers have ample time to notice the subtle shifts in the pendulum's swing plane.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. Earth's gravity exerts a force on objects which gives them potential energy relative to their distance above the ground. This can be calculated with the formula \(U = mgh\), where \(m\) is the mass of the object, \(g\) is the acceleration due to gravity, and \(h\) is the height above the reference point, which is often the lowest point in the swing of a pendulum.

For the Foucault pendulum example, the maximum GPE is reached when the pendulum is at the peak of its swing. It's important to note that as the pendulum swings down and gains speed, this potential energy is converted into kinetic energy, and vice versa, without any loss or gain in total mechanical energy, assuming no external forces such as air resistance are acting on the system. This highlights the interplay between height and energy in gravitational systems and how objects exploit this relationship during oscillatory motion.

Conservation of Mechanical Energy

The principle of conservation of mechanical energy states that in the absence of non-conservative forces, such as friction or air resistance, the total mechanical energy of an isolated system remains constant. Mechanical energy is the sum of potential energy and kinetic energy.

For a swinging pendulum, this principle is beautifully demonstrated. As the pendulum reaches its highest point, its kinetic energy is momentarily zero, and all its mechanical energy is in the form of gravitational potential energy. As it swings down, this energy is converted into kinetic energy - the energy of motion - without any loss. The pendulum has maximum kinetic energy as it passes through the lowest point of its swing, coinciding with the minimum potential energy.

In the absence of non-conservative forces, the maximum gravitational potential energy the pendulum achieves at the highest point equals its maximum kinetic energy at the lowest point. As shown in the exercise's solved example, the maximum kinetic energy calculated via the pendulum's maximum height is identical to the kinetic energy inferred from its swinging speed, providing empirical evidence for the conservation of mechanical energy.