A Foucault pendulum displayed in a museum is typically quite long, making the effect easier to see. Consider a Foucault pendulum of length \(15 \mathrm{~m}\) with a 110 -kg brass bob. It is set to swing with an amplitude of \(3.5^{\circ}\) 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?

The period of a pendulum is a fundamental concept in physics that relates to the time it takes for a pendulum to complete one full swing back and forth. For a simple pendulum, the period, denoted as T, is determined by the formula

\[\begin{equation}T = 2\pi \sqrt{\frac{L}{g}}\end{equation}\]
,where L is the length of the pendulum and g is the acceleration due to gravity. It's fascinating to observe that the period is independent of the mass of the pendulum and its amplitude (as long as the amplitude is small), which means regardless of how heavy the pendulum is or how far you pull it back, it will always take the same amount of time to swing.

The Foucault pendulum exercise illustrates this nicely. Given a pendulum of length 15 meters, the calculated period is approximately 7.74 seconds, meaning it takes just over 7 and a half seconds for the pendulum to swing from one side to the other and back again. This characteristic time is crucial for understanding the rhythmic nature of pendulum motion, and further underscores the elegance and simplicity of the physics governing pendulums.

Kinetic energy, often symbolized as K, is the energy that an object possesses due to its motion. In the context of a swinging pendulum, the kinetic energy is maximal when the pendulum reaches its lowest point during its swing.

Mathematically Defined

It is calculated using the equation

\[\begin{equation}K = \frac{1}{2}mv^2\end{equation}\]
,where m is the mass of the object, and v is its velocity. When dealing with a pendulum, like in the Foucault pendulum we're discussing, evaluating the kinetic energy can be something of a conceptual carousel.

In the example, the maximum kinetic energy is derived from the pendulum's potential energy at its highest point, since energy in a closed system is conserved. With a mass of 110 kg and incorporating gravitational force, we calculate the energy to be approximately 140.71 Joules. This translation of energy types—from potential to kinetic—is a marvelous illustration of the conservation of energy in a closed system and illuminates the underlying physics of pendulum motion.

The conservation of mechanical energy is a powerful principle in physics stating that in the absence of non-conservative forces, such as friction, the total mechanical energy of an isolated system remains constant. This principle combines two types of energy: potential energy and kinetic energy.

For the swinging pendulum in our example, when it reaches the apex of its path, it has maximum potential energy and zero kinetic energy. As it descends, this potential energy is converted into kinetic energy, reaching a maximum at the lowest point of its swing, as we calculated previously.

Main Takeaways

The fascinating aspect of this principle is seen in our Foucault pendulum example, where all the potential energy attributed to its height of 15 meters and mass of 110 kg converts to kinetic energy as it swings down, allowing us to determine its maximum speed. This conversion underscores the pendulum's graceful dance between energy states and is fundamental in understanding not only pendulums but also other mechanical systems where energy transformation is key. The conservation principle ensures that, assuming no energy is lost to external forces, the energetic 'account balance' of the pendulum remains constant throughout its motion.