We use the same initial conditions.ĭescribe the pendulum motion corresponding to Pendulum Trajectory 3. Now we change b to 0.2 (and leave the other coefficients alone).Here is the trajectory corresponding to the same coefficients as in Step 1, but now with initial conditions u 0 = -3, and v 0 = 5.ĭescribe the pendulum motion corresponding to Pendulum Trajectory 2.(b) For each of the points A - F on the trajectory, describe the motion of the pendulum at the corresponding time. (a) In what direction is the trajectory being traced out? To the coordinates of points on a trajectory. This enables us to give a straightforward interpretation On the horizontal axis and its derivative (in our case v = d /DT) OneĪdvantage to the approach we have used is that the phase plane plots the dependent Of defining the new dependent variables u and v when creatingĪ system corresponding to the original second-order differential equation. We use numerical methods to plot the direction field and the trajectory in the phase plane. We'll begin with b = 0 (no damping), u 0 = -3, and v 0 = 0 (zero initial velocity). We'll use units of meters and seconds, so we may take g = 9.807. This is important since the presence of the sine in the equation makes it impossible for us to obtain a symbolic description of the solution. Now we can apply our numerical methods for a system to describe the solution of this problem. With initial conditions u(0) = 0 and v(0) = v 0. Values so that the new initial value problem is equivalent to the original problem. We will construct a system of first-orderĮquations in the dependent variables u and v and assign initial So suppose we have an initial value problem consisting of the pendulum differential equation together with initial conditions However, for our immediate purposes all we need is the differential equation itself. To see more detail on the derivation click here. The statement itself is a consequence of Newton's Second Law of Motion. Here is the angle the pendulum has moved from the vertical, L is the length of the pendulum, g is the acceleration due to gravity, m is the mass of the pendulum, and b is a damping coefficient. In order to have a particular example, let's consider the second-order equation describing the motion of of a pendulum. Angular acceleration / displacement / frequency / velocity Scientists Physics portal Category v t e A pendulum is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity. In particular, it enables us to use first-order numerical methods to approximate the solution of a second-order initial value problem. Such nonlinear differential equations could arise in diverse fields, such as acoustic vibrations, oscillations in small molecules, turbulence and electronic filters, among others.Systems of Differential Equations: Models of Species InteractionĪppendix: Second-Order Differential Equations as Systems Every second-order differential equation may be considered as a system of two first-order equations. This approach may also be profitably used by specialists who encounter during their investigations nonlinear differential equations similar in form to the pendulum equation. Pendulum Equations A simple pendulum consists of a single point of mass m (bob) attached to a rod (or wire) of length and of negligible weight. The treatment is intended for graduate students, who have acquired some familiarity with the hypergeometric functions. We also compare the relative difference between T(0) and T(θ 0) found from the exact equation of motion with the usual perturbation theory estimate. The exact expressions thus obtained are used to plot the graphs that compare the exact time period T(θ 0) with the time period T(0) (based on simple harmonic approximation). The time period of such a pendulum is also exactly expressible in terms of hypergeometric functions. A new and exact expression for the time of swinging of a simple pendulum from the vertical position to an arbitrary angular position θ is given by equation (3.10). In this paper, we provide the exact equation of motion of a simple pendulum of arbitrary amplitude. By using generalized hypergeometric functions, it is however possible to solve the problem exactly. The motion of a simple pendulum of arbitrary amplitude is usually treated by approximate methods.
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