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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

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

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4.2.7 a,b,c
Page Order 56 5 S 5 > Q Q Search Chapter 4. Second-Order Equations underdamped, or undamped? (b) Find a general solution to 0"(t) + ¾0 (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? xercise 4.2.8 An RLC circuit has inductance L = 10-4 henries, resistance R = 0.1 ohms, nd capacitance C= 10-4 farads, with no voltage source so y(t)-0 volts. At time t Othe
Transcribed Image Text:Page Order 56 5 S 5 > Q Q Search Chapter 4. Second-Order Equations underdamped, or undamped? (b) Find a general solution to 0"(t) + ¾0 (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? xercise 4.2.8 An RLC circuit has inductance L = 10-4 henries, resistance R = 0.1 ohms, nd capacitance C= 10-4 farads, with no voltage source so y(t)-0 volts. At time t Othe
Exercise 4.2.7 Consider a pendulum of length L that swings back and forth without friction. Let (r) 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" (1) + 20 (1) = 0 that the function 8 (t) approximately satisfies, at least if the angle (t) remains relatively close to zero (say, 10 (1)| ≤/6, about 30 degrees). (a) Which of the spring-mass models does this correspond to-overdamped, critically damped, 156 Chapter 4. Second-Order Equations
Transcribed Image Text:Exercise 4.2.7 Consider a pendulum of length L that swings back and forth without friction. Let (r) 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" (1) + 20 (1) = 0 that the function 8 (t) approximately satisfies, at least if the angle (t) remains relatively close to zero (say, 10 (1)| ≤/6, about 30 degrees). (a) Which of the spring-mass models does this correspond to-overdamped, critically damped, 156 Chapter 4. Second-Order Equations
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