Problem 137 - Calculus 3

Question

The motion of a swinging pendulum for small angles \(\theta\) can be approximated by \(\frac{d^{2} \theta}{d t^{2}}+\frac{g}{L} \theta=0, \quad\) where \(\theta\) is the angle the pendulum makes with respect to a vertical line, \(g\) is the acceleration resulting from gravity, and \(L\) is the length of the pendulum. Find the equation describing the angle of the pendulum at time \(t,\) assuming an initial displacement of \(\theta_{0}\) and an initial velocity of zero.

Step-by-Step Solution

Answer

\( \theta(t) = \theta_0 \cos\left(\sqrt{\frac{g}{L}}t\right) \)

Step 1: Set Up the Differential Equation

We start with the second-order linear homogeneous differential equation: \( \frac{d^2 \theta}{dt^2} + \frac{g}{L} \theta = 0 \). This represents simple harmonic motion, similar to the equation for a mass-spring system.

Step 2: Recognize the Simple Harmonic Motion

The given equation has a standard form similar to that of a mass on a spring, which allows us to use a general solution for simple harmonic motion: \( \theta(t) = A \cos(\omega t) + B \sin(\omega t) \), where \( \omega^2 = \frac{g}{L} \).

Step 3: Calculate Angular Frequency

Identify \( \omega \), which is the angular frequency for the motion: \( \omega = \sqrt{\frac{g}{L}} \).

Step 4: Apply Initial Conditions to Determine Constants

The initial displacement condition is \( \theta(0) = \theta_0 \). Using the general solution, \( \theta(0) = A \cos(0) + B \sin(0) = A \), so \( A = \theta_0 \).

Step 5: Use Zero Initial Velocity Condition

The initial velocity condition is that \( \frac{d\theta}{dt}(0) = 0 \). Differentiating \( \theta(t) \), we have \( \frac{d\theta}{dt} = -A \omega \sin(\omega t) + B \omega \cos(\omega t) \). At \( t = 0 \), this becomes \( 0 = B \omega \), leading to \( B = 0 \).

Step 6: Write the Solution

Substitute the values of \( A \) and \( B \) back into the equation to get \( \theta(t) = \theta_0 \cos(\omega t) \). By substituting \( \omega = \sqrt{\frac{g}{L}} \), the solution becomes \( \theta(t) = \theta_0 \cos\left(\sqrt{\frac{g}{L}}t\right) \).

Key Concepts

Harmonic Motion

Harmonic motion is a type of periodic motion where an object moves back and forth, or oscillates, through a central equilibrium position. For the case of a pendulum swinging in small angles, the motion can be approximated as "simple harmonic motion" due to its sinusoidal nature. This kind of motion can be described mathematically using sinusoidal functions such as sine and cosine.

When an object exhibits harmonic motion, it implies that its restoring force is directly proportional to its displacement but in the opposite direction. This is evident in the pendulum swinging gently due to gravitational force. The pendulum, like a mass on a spring, returns to its original position with a predictable period and frequency. Harmonic motion in pendulums is beautifully simple, allowing us to use fundamental dynamics to predict future behavior effectively.

Differential Equations

Differential equations are mathematical equations that involve derivatives, which represent rates of change. These equations are crucial in modeling how quantities change over time or space. In the context of pendulum motion, the differential equation for a simple pendulum is expressed as \(\frac{d^{2} \theta}{d t^{2}}+\frac{g}{L} \theta=0\). This second-order linear homogeneous differential equation captures the essence of motion under oscillations.

Solving this equation involves finding a function \(\theta(t)\) representing the angle over time. Recognizing the similarity of this equation to other harmonic systems, like mass-spring systems, allows us to leverage known solutions. The result is a sinusoidal function that describes the motion. By solving differential equations, one can predict the behavior of dynamic systems, such as the swinging of a pendulum, based on initial conditions and parameters like gravity \(g\) and pendulum length \(L\).

Angular Frequency

Angular frequency, denoted by \(\omega\), is a measure of how quickly an object goes through its cycles in oscillatory motion. It is linked to the concept of frequency, which is the number of cycles per unit time. For pendulum motion, the angular frequency can be calculated from the parameters of the system, specifically the gravitational acceleration \(g\) and the length of the pendulum \(L\).

The angular frequency is derived using the formula \(\omega = \sqrt{\frac{g}{L}}\). This expression emerges naturally by recognizing the structure of the differential equation for pendulum motion. Angular frequency is instrumental in determining the period of oscillation, as it defines how fast the pendulum swings back and forth. A larger angular frequency indicates a quicker oscillation, while a smaller one suggests slower movement.

Initial Conditions

Initial conditions are the values or constraints set at the beginning of a problem to determine a unique solution to a differential equation. In analyzing pendulum motion, initial conditions are crucial in defining the characteristics of the solution curve. For a pendulum, they typically include initial displacement \(\theta_0\) and initial velocity.

For the given problem, the pendulum starts with an initial angle \(\theta_0\) and zero initial velocity. These conditions allow us to find specific constants in the general solution function \(\theta(t) = A \cos(\omega t) + B \sin(\omega t)\). When applying the initial displacement, \(A\) equals the initial angle, and using initial velocity, we find \(B = 0\). Therefore, these conditions specify the motion of the pendulum uniquely, resulting in the equation \(\theta(t) = \theta_0 \cos\left(\sqrt{\frac{g}{L}}t\right)\). Understanding initial conditions helps in anchoring the theoretical model to the physical scenario, providing precise predictions and insights into pendulum dynamics.