Question

Suppose that at some initial time the pendulum is located at angle \(\theta_{0}\) with an angular velocity \(d \theta / d t=\omega_{0}\) radians/sec. (a) Equation (15) is a second order differential equation. Rewrite it as a first order separable equation by adopting angle \(\theta\) as the independent variable, using the fact that $$ \theta^{\prime \prime}=\frac{d}{d t}\left(\frac{d \theta}{d t}\right)=\frac{d \omega}{d t}=\frac{d \omega}{d \theta} \frac{d \theta}{d t}=\omega \frac{d \omega}{d \theta} . $$ Complete the specification of the initial value problem by specifying an appropriate initial condition. (b) Obtain the implicit solution $$ m l^{2} \frac{\omega^{2}}{2}-m g l \cos \theta=m l^{2} \frac{\omega_{0}^{2}}{2}-m g l \cos \theta_{0^{-}} $$ The pendulum is a conservative system; that is, energy is neither created nor destroyed. Equation (16) is a statement of conservation of energy. At a position defined by the angle \(\theta\), the quantity \(m l^{2} \omega^{2} / 2\) is the kinetic energy of the pendulum while the term \(-m g l \cos \theta\) is the potential energy, referenced to the horizontal position \(\theta=\pi / 2\). The constant right-hand side is the total initial energy. (c) Determine the angular velocity at the instant the pendulum reaches the vertically downward position, \(\theta=0\). Express your answer in terms of the constants \(\omega_{0}\) and \(\theta_{0}\).

Short answer

(a) Rewrite the second-order differential equation as a first-order separable equation and provide the initial condition. The first-order separable equation is given by: $$\frac{d\omega}{d\theta}=\frac{\theta^{\prime \prime}}{\omega}$$ with initial condition $$\omega(\theta_0) = \omega_0.$$ (b) The given implicit solution represents the conservation of energy in the pendulum system. (c) The angular velocity of the pendulum when it reaches the vertically downward position (\(\theta=0\)) is given by: $$\omega=\sqrt{\omega_{0}^2 -2\frac{g}{l}(1-\cos\theta_{0})}.$$

Step-by-step solution

Rewrite as first-order separable equation

We are given the relationship: $$\theta^{\prime \prime}=\frac{d}{d t}\left(\frac{d \theta}{d t}\right)=\frac{d \omega}{d t}=\frac{d \omega}{d \theta} \frac{d \theta}{d t}=\omega \frac{d \omega}{d \theta}.$$ We want to rewrite it as a first-order separable equation \(\frac{d\omega}{d\theta}=f(\theta,\omega)\). We're already given the fact that: $$\frac{d\omega}{d\theta}=\frac{\theta^{\prime \prime}}{\omega}$$ Now, we specify an appropriate initial condition: At the initial time, the angular position is \(\theta_{0}\) and the angular velocity is \(\omega_{0}\). Thus, the initial condition is: $$\omega(\theta_{0})=\omega_{0}.$$

Obtain the implicit solution

The implicit solution given is: $$m l^{2} \frac{\omega^{2}}{2}-m g l \cos \theta=m l^{2}\frac{\omega_{0}^{2}}{2}-m g l \cos \theta_{0}.$$ This equation represents the conservation of energy; the left side of the equation accounts for the kinetic and potential energies of the pendulum at angle \(\theta\), while the constant right side represents the total initial energy of the system.

Determine angular velocity at \(\theta=0\)

We need to find \(\omega\) at the instant the pendulum reaches the vertically downward position \(\theta=0\). Substitute \(\theta=0\) into the implicit solution and solve for \(\omega\): $$m l^{2} \frac{\omega^{2}}{2}-m g l \cos 0=m l^{2}\frac{\omega_{0}^{2}}{2}-m g l \cos \theta_{0}.$$ Since \(\cos 0 = 1\), we have: $$m l^{2} \frac{\omega^{2}}{2}-m g l = m l^{2}\frac{\omega_{0}^{2}}{2}-m g l \cos \theta_{0}.$$ Now we solve for \(\omega^2\): $$\omega^2 = \omega_{0}^2 -2\frac{g}{l}(1-\cos\theta_{0}).$$ Thus, the angular velocity when the pendulum reaches the vertically downward position is: $$\omega=\sqrt{\omega_{0}^2 -2\frac{g}{l}(1-\cos\theta_{0})}.$$

Key concepts

First Order Separable Equations

When dealing with differential equations, a first order separable equation is one that can be restructured into a form where each side of the equation consists of a function of one variable only. This is the kind of equation that students encounter when analyzing the dynamics of a pendulum, as seen in the exercise.

For a pendulum, by considering the angular velocity \(\omega\) as a function of the angle \(\theta\), we can write the equation of motion in a separable form. This is accomplished by taking the second derivative of \(\theta\) with respect to time (which is the angular acceleration) and expressing it in terms of \(\theta\) and \(\omega\). Essentially, you manipulate the equation such that one side has differentials of \(\theta\) and the other \(\omega\).

In this way, the process of solving the differential equation is transformed into isolating the variables and integrating both sides, allowing for the determination of \(\omega\) as a function of \(\theta\). Importantly, initial conditions, like the angle and angular velocity at the start of the pendulum's motion, are used to fully determine the solution.

Conservation of Energy

The principle of conservation of energy is a fundamental concept in physics which declares that energy in a closed system remains constant—it is neither created nor destroyed, but may transform from one form to another.

In the context of a pendulum, this principle plays a crucial role. The equation derived in the exercise, \[ m l^{2} \frac{\omega^{2}}{2}-m g l \cos \theta = m l^{2} \frac{\omega_{0}^{2}}{2}-m g l \cos \theta_{0} \], encapsulates this principle perfectly by equating kinetic energy (associated with angular velocity, \(\omega\)) and potential energy (associated with the height of the pendulum, thus the angle \(\theta\)) at any moment to the total mechanical energy which remains constant over time.

Understanding how to apply this conservation law allows students to solve for unknown quantities, such as the angular velocity at different positions, without having to directly solve the differential equation governing the pendulum's motion. It's an incredibly powerful tool that simplifies complex problems into manageable calculations.

Angular Velocity

Angular velocity, symbolized as \(\omega\), defines the rate at which an object rotates or revolves about a point or axis. In the case of a pendulum, it's the speed at which the pendulum swings through different angles, measured in radians per second.

The exercise provides a practical application of angular velocity: determining its value when the pendulum reaches the lowest point in its swing (\(\theta=0\)). At this position, the pendulum has the maximum kinetic energy, as all the potential energy has been converted into kinetic during the swing. From the conservation of energy equation derived in the problem, we can isolate and solve for \(\omega\) at that instance, providing a tangible insight into how angular velocity varies with motion.

Having a grasp on how to calculate angular velocity not only furthers the understanding of pendular motion but also enriches the broader comprehension of rotational dynamics, crucial in various branches of science and engineering.