Question

Write down the potential energy \(U(\phi)\) of a simple pendulum (mass \(m,\) length \(l\) ) in terms of the angle \(\phi\) between the pendulum and the vertical. (Choose the zero of \(U\) at the bottom.) Show that, for small angles, \(U\) has the Hooke's law form \(U(\phi)=\frac{1}{2} k \phi^{2},\) in terms of the coordinate \(\phi .\) What is \(k ?\)

Short answer

For small angles, the potential energy is \( U(\phi) = \frac{1}{2} mg l \phi^2 \). Here, \( k = mg l \).

Step-by-step solution

Understanding the Setup of the Pendulum

In this problem, we need to calculate the potential energy of a pendulum, which is a mass attached to a rod of length \( l \). The potential energy depends on the change in height of the pendulum, which in turn depends on the angle \( \phi \) the pendulum makes with the vertical.

Expressing Height in Terms of Angle

To find the potential energy, we first express the change in height \( h \) as a function of the angle \( \phi \). When the pendulum is displaced by the angle \( \phi \), its height change relative to the lowest point is \[ h = l (1 - \cos(\phi)). \]

Writing the Potential Energy Function

The potential energy \( U \) is given by \( U = mgh \), where \( g \) is the acceleration due to gravity. Substituting \( h \) gives \[ U(\phi) = mg l (1 - \cos(\phi)). \] This is the potential energy as a function of the angle \( \phi \).

Approximating for Small Angles

For small angles \( \phi \), we use the approximation \( \cos(\phi) \approx 1 - \frac{\phi^2}{2} \). Substituting this into the potential energy formula, we have \[ U(\phi) \approx mg l \left(1 - \left(1 - \frac{\phi^2}{2}\right)\right) = \frac{1}{2} mg l \phi^2. \]

Identifying Hooke's Law Form and Determining \( k \)

The expression \( \frac{1}{2} mg l \phi^2 \) resembles the Hooke's Law form \( U(\phi) = \frac{1}{2} k \phi^2 \), where \( k \) is the effective spring constant. By comparing, we find that \( k = mg l \).

Key concepts

Hooke's Law

Hooke's Law is fundamental to understanding the behavior of springs and similar systems where restoring forces are present. It states that the force needed to extend or compress a spring by some distance is proportional to that distance. Mathematically, it's expressed as \[ F = -kx \]where:- \( F \) is the force exerted by the spring,- \( k \) is the spring constant, and- \( x \) is the displacement from the equilibrium position.
When thinking about a simple pendulum, like the one in our exercise, the concept becomes a little more abstract.
Instead of a spring, we have a mass swinging back and forth. As it swings, gravity works to pull it back toward its lowest point.For small displacements, this behavior can be likened to a spring, with the potential energy having a quadratic form just like Hooke's Law. In our case, the spring constant \( k \) is effectively replaced by \( mg l \), connecting the pendulum's potential energy to Hooke's Law.

Small Angle Approximation

The small angle approximation is a handy tool in physics that simplifies the mathematics involved in analyzing systems like pendulums.
When dealing with angles generally measured in radians, small angles (those less than around 10 degrees) are approximated to streamline calculations.
For a pendulum, when the angle \( \phi \) is small, we make the approximation \[ \cos(\phi) \approx 1 - \frac{\phi^2}{2}. \]This means the series expansion of cosine can be truncated for small \( \phi \) values, allowing us to simplify expressions related to the pendulum's motion and potential energy.

• This approximation is incredibly useful for simplifying the potential energy expression,
• which in turn makes it easier to relate to Hooke's Law.
• It is critical for the mathematical analysis of the pendulum to use this approximation when angles are small.

Pendulum Dynamics

The dynamics of a simple pendulum involve its motion under the influence of gravitational force. When a pendulum swings, several forces interact:

• The tension in the rod or string keeps the mass moving in a circular arc.
• Gravity pulls the mass downwards, creating a potential energy depending on its height.
• The angle \( \phi \) determines both the position and height.

For small angles, these dynamics resemble the behavior of a harmonic oscillator as described by Hooke's Law.
The pendulum dynamics are simpler with the small angle approximation, as it linearizes the equations of motion.This linearization means the pendulum's restoring force and potential energy can be modeled similarly to a spring, facilitating predictions about its motion and energy transformations.