Problem 20 - Differential Equations and Linear Algebra

Question

Deal with the simple pendulum whose motion is described by Equation ( 8.5 .28 ). Show that the period of the simple pendulum is \(T=\) \(2 \pi \sqrt{L / g} .\) Determine the length of a pendulum that takes one second to swing from its extreme position on the right to its extreme position on the left. Let \(g=9.8\) meters/second \(^{2}\)

Step-by-Step Solution

Answer

In summary, the period of a simple pendulum is given by \(T=2\pi\sqrt{\frac{L}{g}}\), where \(L\) is the length of the pendulum and \(g\) is the acceleration due to gravity. For a pendulum with a one-second period and the given value of \(g=9.8 \frac{\text{m}}{\text{s}^2}\), the length is approximately 0.248 meters.

Step 1: Equation of motion for a simple pendulum

A simple pendulum consists of a point mass, called the bob, attached to a massless, inextensible string of length L. When the pendulum is displaced from its equilibrium position, it experiences a restoring force, which is the component of gravity acting along the arc of motion. To find the equation of motion for a simple pendulum, let's consider the following variables: - θ(t): the angle between the string and the vertical axis at time t - T: tension in the string - m: mass of the bob When the pendulum is at an angle θ(t), the force analysis on the bob gives us the following equation: \[ m\frac{d^2\theta}{dt^2}=-mg\sin\theta \] Now, we have the equation of motion of a simple pendulum as \[\frac{d^2\theta}{dt^2}=-\frac{g}{L}\sin\theta\]

Step 2: Simplify the equation of motion using small-angle approximation

For small angles (θ close to 0), we can use the small-angle approximation: \[\sin\theta \approx \theta\] Applying this approximation, the equation of motion becomes: \[\frac{d^2\theta}{dt^2}=-\frac{g}{L}\theta\]

Step 3: Solve for the period of the simple pendulum

The equation of motion is now a simple harmonic oscillator equation. To find the period, we can analyze the angular frequency (ω), which is related to the coefficients of the formula: \[\omega^2=\frac{g}{L}\] Taking the square root of both sides: \[\omega=\sqrt{\frac{g}{L}}\] The period (T) is the inverse of the angular frequency: \[T=\frac{2\pi}{\omega}\] Now, substitute the ω expression: \[T=2\pi\sqrt{\frac{L}{g}}\] This equation gives us the period of a simple pendulum.

Step 4: Determine the length of a pendulum with a one-second period

We need to find a pendulum length (L) that results in a one-second period (T). Given \(g=9.8 \frac{\text{m}}{\text{s}^2}\), we can use the period equation to solve for L: \[1 = 2\pi\sqrt{\frac{L}{9.8}}\] First, square both sides of the equation: \[1^2 = \left(2\pi\sqrt{\frac{L}{9.8}}\right)^2\] Next, solve for L: \[L = \frac{9.8}{\left(2\pi\right)^2}\] Finally, calculate the value for L: \[L \approx 0.248\,\text{m}\] So, the length of a pendulum that takes one second to swing from one extreme position to another is approximately 0.248 meters.

Key Concepts

Equation of Motion

The equation of motion for a simple pendulum describes how the pendulum's position changes over time. A simple pendulum consists of a point mass (the bob) attached to a string that can swing freely. The string is inextensible and massless, meaning it doesn't stretch and doesn't contribute to the overall mass. When the pendulum is displaced from its resting (equilibrium) position, it experiences a restoring force. This force is the result of gravity acting on the mass of the pendulum.

The angle between the pendulum and the vertical at any time is denoted by \( \theta(t) \). The restoring force provides a torque that tends to pull the pendulum back to the equilibrium position. The forces at play lead to the following differential equation:

• \( m\frac{d^2\theta}{dt^2} = -mg\sin\theta \)

Here, \( m \) is the mass of the pendulum bob, \( g \) is the acceleration due to gravity, and \( \sin\theta \) is the sine of the angle. This equation governs the motion of a simple pendulum and shows that its motion is determined by gravity and the length of the string.

Period of Pendulum

The period of a pendulum refers to the time it takes for the pendulum to make one complete back-and-forth swing. Understanding this is crucial, as it indicates how fast or slow a pendulum will move over time. To find the period of the simple pendulum, we first need to rewrite the equation of motion using a well-known physics approximation.

The formula for the period of a simple pendulum, derived from harmonics, is given as:

• \( T = 2\pi\sqrt{\frac{L}{g}} \)

This equation shows that the period \( T \) depends on two factors:

• Length of the string \( L \)
• Acceleration due to gravity \( g \)

We see that the period is independent of the mass of the bob and only depends on the pendulum's length and gravity. This fundamental property is significant in designing pendulum-based timekeeping devices, like clocks, because it allows for uniform time intervals.

Small-Angle Approximation

When analyzing the motion of a simple pendulum, if the oscillations are very small, i.e., \( \theta \) is close to zero, we can simplify the problem using the small-angle approximation. In this approximation, we assume:

• \( \sin\theta \approx \theta \)

This simplification turns the complex trigonometric component into a manageable linear term. The equation of motion, therefore, simplifies from:

• \( \frac{d^2\theta}{dt^2} = -\frac{g}{L}\sin\theta \)

to

• \( \frac{d^2\theta}{dt^2} = -\frac{g}{L}\theta \)

This linear form corresponds to the differential equation of simple harmonic motion. This approximation works exceptionally well for small angles, leading to predictable and regular motion patterns. The simplification is a key step in finding the pendulum's period and explaining the pendulum as a classic example of a harmonic oscillator.

Harmonic Oscillator

A harmonic oscillator refers to any system that, when displaced from its equilibrium, experiences a restoring force proportional to that displacement. Many systems in physics, including pendulums at small angles, exhibit this type of motion. For our pendulum using the small-angle approximation, the equation resembles that of a harmonic oscillator.

As a result, the system follows the rules of simple harmonic motion:

• The acceleration is directly proportional to the negative of the displacement.
• The motion is periodic and follows a sine or cosine curve.

Because the pendulum matches these criteria under the small-angle approximation, its behavior is predictable. The linearized equation implies that the angular frequency \( \omega \) of the system is \( \omega = \sqrt{\frac{g}{L}} \). The equation for the period, \( T = \frac{2\pi}{\omega} \), confirms that the pendulum's motion is periodic and cyclical.

This understanding of pendulums as harmonic oscillators is vital, as it's a model for various natural and mechanical systems. Observing these principles in a pendulum provides insight into more complex oscillatory systems throughout physics and engineering.