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Exercise 4.2.7 Consider a pendulum of length L that swings back and forth without friction. Let 8(t) be the angle that the pendulum makes with a vertical line; see Figure 4.32 and Sections 4.6.5 or 4.6.6, where we derive the ODE 8" (1) + 0 (t) = 0 that the function 8 (t) approximately satisfies, at least if the angle (t) remains relatively close to zero (say, 8(t)| ≤ π/6, about 30 degrees). (a) Which of the spring-mass models does this correspond to overdamped, critically damped,

Exercise 4.2.7 Consider a pendulum of length L that swings back and forth without friction. Let 8(t) be the angle that the pendulum makes with a vertical line; see Figure 4.32 and Sections 4.6.5 or 4.6.6, where we derive the ODE 8" (1) + 0 (t) = 0 that the function 8 (t) approximately satisfies, at least if the angle (t) remains relatively close to zero (say, 8(t)| ≤ π/6, about 30 degrees). (a) Which of the spring-mass models does this correspond to overdamped, critically damped,

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Exercise 4.2.7 Consider a pendulum of length L that swings back and forth without friction. Let 8(t) be the angle that the pendulum makes with a vertical line; see Figure 4.32 and Sections 4.6.5 or 4.6.6, where we derive the ODE 0" (t) + 20 (1) = 0 that the function (t) approximately satisfies, at least if the angle (t) remains relatively close to zero (say, 0(t)| ≤ π/6, about 30 degrees). (a) Which of the spring-mass models does this correspond to- Ove amped, critically damped, 156 Chapter 4. Second-Order Equations underdamped, or undamped? (b) Find a general solution to 0"(t) + 0 (t) = 0. (c) Find a formula for P, the period of the pendulum (one back and forth swing) in terms of g and L. Do a quick check on the reasonableness of your formula-what does it predict if L is larger or smaller? What if g were larger or smaller?
Transcribed Image Text:Exercise 4.2.7 Consider a pendulum of length L that swings back and forth without friction. Let 8(t) be the angle that the pendulum makes with a vertical line; see Figure 4.32 and Sections 4.6.5 or 4.6.6, where we derive the ODE 0" (t) + 20 (1) = 0 that the function (t) approximately satisfies, at least if the angle (t) remains relatively close to zero (say, 0(t)| ≤ π/6, about 30 degrees). (a) Which of the spring-mass models does this correspond to- Ove amped, critically damped, 156 Chapter 4. Second-Order Equations underdamped, or undamped? (b) Find a general solution to 0"(t) + 0 (t) = 0. (c) Find a formula for P, the period of the pendulum (one back and forth swing) in terms of g and L. Do a quick check on the reasonableness of your formula-what does it predict if L is larger or smaller? What if g were larger or smaller?
4.6.5 Project: The Pendulum In this project we derive the equation of motion for a pendulum using conservation of energy. In the project "The Pendulum 2" of Section 4.6.6 you can explore an alternate derivation based on Newton's second law of motion; that derivation also incorporates friction, while this one does not. Consider a pendulum of length L as depicted in Figure 4.32. As the pendulum swings back and forth it makes an angle 0 (t) with respect to vertical at time t. In this exercise we will derive two nonlinear differential equations that govern the pendulum's motion, one first-order, the other second-order, using a simple conservation of energy argument. We will then approximate the resulting nonlinear ODEs with a simpler linear second-order ODE, and compare the solutions. (0,L) DEC Lcos (0 (t)) e(t) L (0,0) Figure 4.32: A pendulum of length L at an angle of 0(t) with respect to the vertical. Let us take the position of the pendulum's pivot as the point (0,L) in a standard xy coordinate system. Assume that the bob (the mass at the end of the pendulum) has mass m, and that the thin rod that connects the bob to the pivot has negligible mass. The xy position of the bob at any time is tv logical; from examination of a single family to comparative assess- A I 2 02 20
Transcribed Image Text:4.6.5 Project: The Pendulum In this project we derive the equation of motion for a pendulum using conservation of energy. In the project "The Pendulum 2" of Section 4.6.6 you can explore an alternate derivation based on Newton's second law of motion; that derivation also incorporates friction, while this one does not. Consider a pendulum of length L as depicted in Figure 4.32. As the pendulum swings back and forth it makes an angle 0 (t) with respect to vertical at time t. In this exercise we will derive two nonlinear differential equations that govern the pendulum's motion, one first-order, the other second-order, using a simple conservation of energy argument. We will then approximate the resulting nonlinear ODEs with a simpler linear second-order ODE, and compare the solutions. (0,L) DEC Lcos (0 (t)) e(t) L (0,0) Figure 4.32: A pendulum of length L at an angle of 0(t) with respect to the vertical. Let us take the position of the pendulum's pivot as the point (0,L) in a standard xy coordinate system. Assume that the bob (the mass at the end of the pendulum) has mass m, and that the thin rod that connects the bob to the pivot has negligible mass. The xy position of the bob at any time is tv logical; from examination of a single family to comparative assess- A I 2 02 20
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