Question

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?

Short answer

Answer: The period of the pendulum is approximately 7.74 seconds, the maximum kinetic energy is about 140.71 Joules, and the maximum speed is approximately 1.47 m/s.

Step-by-step solution

Using the formula for the period of a pendulum

The formula to find the period of a pendulum is \(T = 2\pi \sqrt{\frac{L}{g}}\), where \(T\) is the period, \(L\) is the length of the pendulum, and \(g\) is the acceleration due to gravity, which is approximately \(9.81 \mathrm{~m/s^2}\). Plugging in the given length \(L=15\mathrm{~m}\), we can calculate the period.$$ T = 2\pi \sqrt{\frac{15\mathrm{~m}}{9.81\mathrm{~m/s^2}}} $$

Computing the period

Now, we can solve for the period:$$ T = 2\pi \sqrt{\frac{15\mathrm{~m}}{9.81\mathrm{~m/s^2}}} \approx 7.74\mathrm{~s} $$The period of the pendulum is approximately \(7.74\) seconds. #b) Finding the maximum kinetic energy of the pendulum#

Using the conservation of mechanical energy principle

The pendulum's total mechanical energy is conserved, meaning that its potential energy (\(U\)) at the highest point is equal to its kinetic energy (\(K\)) at the lowest point. At the highest point, the pendulum's potential energy can be computed using the formula \(U = mgh\), where \(m\) is the mass of the pendulum, \(h\) is the change in height, and we've already defined \(g\). We can also find the height change using trigonometry and pendulum amplitude.

Calculate the change in height

We can find the change in height by calculating the difference of the vertical component of the initial and final positions of the bob with respect to the equilibrium point. In this case, the initial position is the maximum height, and the final position is at the equilibrium. The change in height can be computed using trigonometric identities:$$ h = L - L \cos{3.5^\circ} $$

Compute the potential energy

Now, we can compute the potential energy at the highest point, which will be equal to the maximum kinetic energy:$$ U = mgh = (110\mathrm{~kg})(9.81\mathrm{~m/s^2})(15\mathrm{~m} - 15\mathrm{~m} \cos{3.5^\circ}) \approx 140.71\mathrm{~J} $$The maximum kinetic energy of the pendulum is approximately \(140.71\) Joules. #c) Finding the maximum speed of the pendulum#

Using the relationship between kinetic energy and velocity

The kinetic energy can be defined as \(K = \frac{1}{2}mv^2\), where \(v\) is the velocity of the pendulum. The maximum kinetic energy that we found in part b is equal to the kinetic energy at the pendulum's lowest point, where it has the maximum velocity. So, we have$$ 140.71\mathrm{~J} = \frac{1}{2}(110\mathrm{~kg})v^2 $$

Solve for the maximum velocity

Now, we can solve for the maximum velocity:$$ v = \sqrt{\frac{2(140.71\mathrm{~J})}{110\mathrm{~kg}}} \approx 1.47\mathrm{~m/s} $$The maximum speed of the pendulum is approximately \(1.47\) \(\mathrm{m/s}\).

Key concepts

Pendulum Period

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

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.

Conservation of Mechanical Energy

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.