The simple pendulum consists of a small bob of mass m that is attached to the end of a string. The pendulum is released from rest when θ = 30°. Using numerical integration of the differential equation of motion, calculate (1) the period of oscillation; and (2) the maximum angular velocity. Compare

Question

The simple pendulum consists of a small bob of mass m that is attached to the end of a string. The pendulum is released from rest when θ = 30°. Using numerical integration of the differential equation of motion, calculate (1) the period of oscillation; and (2) the maximum angular velocity. Compare the maximum angular velocity to the exact value obtained by the work-energy method.

Accepted Answer

Because the pendulum is a conservative system, it oscillates between θ = ±30°. The motion is periodic, but not harmonic, because θ is not a small angle—that is, not less than 6°. We calculate the period of the pendulum by noting that the time taken by the pendulum to travel from the initial position (θ = 30°) to the vertical position (θ = 0) equals one-fourth of the period.
From Eq. (20.9), the differential equation of motion is [latex]\ddot θ[/latex] = −(g/L) sin θ = −(9.81/0.750) sin θ = −13.080 sin θ. The equivalent first-order equations are

[latex]\ddot θ + \frac{g}{L} sin θ = 0[/latex] (20.9)

[latex]\dot θ[/latex] = ω [latex]\dot ω[/latex] = −13.080 sin θ

with the initial conditions θ = 30° = π/6 rad and ω = 0 at t = 0 (the time of release). The integration interval extends from t = 0 until the time when θ = 0 for the first time. An estimate of the period can be obtained from the linearized differential equation, Eq. (20.10): [latex]\ddot θ[/latex]+(g/L)θ = 0. We get from Eq. (20.7) [latex]τ = 2π/p =2π/\sqrt{g/L }= 2π/\sqrt{9.81/0.75}[/latex] = 1.737 s. To be on the safe side, the integration period for the nonlinear problem should be somewhat larger than 1.737/4. We used 0.45 s in the MATLAB program listed below.

[latex]\ddot θ+\frac{g}{L}θ = 0[/latex] (20.10)

[latex]τ = \frac{2π}{p}[/latex] (20.7)

Question 20.2: The simple pendulum consists of a small bob of mass m that i...

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