Use the result of Problem 8 to prove that if the pendulum in...

Use the result of Problem 8 to prove that if the pendulum in Figure 4.18 on page 208 is released from rest at the angle $ \alpha, 0<\alpha<\pi $ then $ |\theta(t)| \leq \alpha $ for all $ t $ [Hint: The initial conditions are $ \theta(0)=\alpha, \theta^{\prime}(0)=0 $; argue that the constant in Problem 8 equals $ -(g / \ell) \cos \alpha $ .]

Key Concepts

Conservation Laws and Integration Constants

By integrating the equation of motion, one obtains a conserved quantity—a constant of integration—that is determined by the initial conditions. This constant captures the interplay between kinetic and potential energies and is critical in deriving inequalities that bound the motion of the pendulum.

Energy Conservation

In mechanical systems, energy conservation is the principle that the total energy—typically the sum of kinetic and potential energy—remains constant over time when only conservative forces are acting. For a pendulum, the initial potential energy given by its displacement sets an upper limit on the kinetic energy it can acquire, which in turn restricts the range of motion.

Nonlinear Oscillatory Motion

The motion of a pendulum is described by a nonlinear differential equation due to the sine function in its restoring force. Analyzing such nonlinear systems often requires techniques that can handle the lack of simple linearity, using integral invariants or phase-plane methods to understand the behavior and bounds of the solution.

Initial Value Problems and Bounded Solutions

The initial conditions of a system, like setting the pendulum's initial angle and zero initial velocity, dictate specific values for integration constants. These constants then guarantee that the solution, such as the pendulum’s angle over time, cannot exceed certain bounds, ensuring that the oscillations remain within a predictable range.

Transcript

00:01 This problem is about the pendulum again.

00:03 And so let's do a quick sketch of what the pendulum is like.

00:07 Okay? so that's the line of equilibrium.

00:10 Let's call this point o.

00:12 And here we have some line.

00:15 That's the length of the pendulum.

00:17 We have the mass that's attached at the end of the pendulum.

00:21 Of course we have some forces here.

00:23 The gravity is pointing down.

00:26 M .g.

00:28 Okay.

00:28 And here we have the tangent of this curve, which is m g sine theta.

00:47 Okay, and this is the angle of the pendulum at some time t, theta of t.

00:55 Now in this problem, it's released from an initial angle of theta.

01:01 So initial angle of theta, of alpha, i'm sorry, within the interval 0 pi.

01:10 Okay, so that's the initial angle that we're released.

01:13 At and so we're just releasing it we're not pushing it and so what we know is that the velocity at time zero is equal to zero and the initial angle at time zero is equal to alpha okay what we'd like to show in the question mark is that the constant in the energy integral lemma is equal to negative g over l, cosine of this initial angle alpha...

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