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The restoring force of a simple pendulum in the angular (0) direction is F = -mg sin 0, and the corresponding acceleration is a = -Folm = -g sin 0. Dividing a by the pendulum length L yields the angular acceleration α = ag/L=-- dQ2 We know from rotational kinematics that angular acceleration is the rate of change of angular velocity, i.e. a = = S2, and the angular velocity is the dt rate of change of the angle 8, i.e. Ω = Therefore, our differential equations of motion for the simple pendulum are dQ dt de dt S2₁ = 20 + de dt = 22 @= £sin dQ dt L sin 0 Why do we use a capital greek omega (2) for the angular velocity? Because the lowercase @w is defined in the lab manual as the angular frequency of the pendulum, which is a different (constant) quantity: -sin 0 Ꮎ 8 VL Iteration is King (again)! The Euler-Cromer method of numerical integration To solve these differential equations, we use (almost) the same interactive Euler method as for the "free fall/air drag" experiment. First, let's define the initial values: We start the pendulum at time to from an angle 80 with zero angular velocity (20 = 0). For a sufficiently short time interval At, the angular acceleration can be considered constant, so we can calculate the new angular velocity at time t₁ = to + At: At = - =sin(00) At L

The restoring force of a simple pendulum in the angular (0) direction is F = -mg sin 0, and the corresponding acceleration is a = -Folm = -g sin 0. Dividing a by the pendulum length L yields the angular acceleration α = ag/L=-- dQ2 We know from rotational kinematics that angular acceleration is the rate of change of angular velocity, i.e. a = = S2, and the angular velocity is the dt rate of change of the angle 8, i.e. Ω = Therefore, our differential equations of motion for the simple pendulum are dQ dt de dt S2₁ = 20 + de dt = 22 @= £sin dQ dt L sin 0 Why do we use a capital greek omega (2) for the angular velocity? Because the lowercase @w is defined in the lab manual as the angular frequency of the pendulum, which is a different (constant) quantity: -sin 0 Ꮎ 8 VL Iteration is King (again)! The Euler-Cromer method of numerical integration To solve these differential equations, we use (almost) the same interactive Euler method as for the "free fall/air drag" experiment. First, let's define the initial values: We start the pendulum at time to from an angle 80 with zero angular velocity (20 = 0). For a sufficiently short time interval At, the angular acceleration can be considered constant, so we can calculate the new angular velocity at time t₁ = to + At: At = - =sin(00) At L

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.3: The Addition And Subtraction Formulas
Problem 72E
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The restoring force of a simple pendulum in the angular (0) direction is Fo = -mg sin 0, and the corresponding acceleration is a = - Folm = -g sin 0. Dividing a by the pendulum length L yields the angular acceleration α = a₂/L dQ dt We know from rotational kinematics that angular acceleration is the rate of change of angular velocity, i.e. a = rate of change of the angle , i.e. Therefore, our differential equations of motion for the simple pendulum are dQ dt do dt Ω = 221 g == sin 0 L Why do we use a capital greek omega () for the angular velocity? Because the lowercase @ is defined in the lab manual as the angular frequency of the pendulum, which is a different (constant) quantity: de dt g == sin 0 We repeat these calculations over and over. For the ith step, we have g --√i @= Iteration is King (again)! The Euler-Cromer method of numerical integration To solve these differential equations, we use (almost) the same interactive Euler method as for the "free fall/air drag" experiment. First, let's define the initial values: We start the pendulum at time to from an angle with zero angular velocity (20 = 0). = dQ = 20 + At dt For a sufficiently short time interval At, the angular acceleration can be considered constant, so we can calculate the new angular velocity at time t₁ = to + At: sin(00) At №₁ = Ω Ω-1 = Likewise, the angular velocity does not change much during At, so the new angle at time t₁ is θι = θα + Ω, ΔΙ t₁ = ti-1 + At Ω and the angular velocity is the N₁1 — — sin(0₁-1)At L 0; = 0;−1 + Ω; ΔΙ i-1
Transcribed Image Text:The restoring force of a simple pendulum in the angular (0) direction is Fo = -mg sin 0, and the corresponding acceleration is a = - Folm = -g sin 0. Dividing a by the pendulum length L yields the angular acceleration α = a₂/L dQ dt We know from rotational kinematics that angular acceleration is the rate of change of angular velocity, i.e. a = rate of change of the angle , i.e. Therefore, our differential equations of motion for the simple pendulum are dQ dt do dt Ω = 221 g == sin 0 L Why do we use a capital greek omega () for the angular velocity? Because the lowercase @ is defined in the lab manual as the angular frequency of the pendulum, which is a different (constant) quantity: de dt g == sin 0 We repeat these calculations over and over. For the ith step, we have g --√i @= Iteration is King (again)! The Euler-Cromer method of numerical integration To solve these differential equations, we use (almost) the same interactive Euler method as for the "free fall/air drag" experiment. First, let's define the initial values: We start the pendulum at time to from an angle with zero angular velocity (20 = 0). = dQ = 20 + At dt For a sufficiently short time interval At, the angular acceleration can be considered constant, so we can calculate the new angular velocity at time t₁ = to + At: sin(00) At №₁ = Ω Ω-1 = Likewise, the angular velocity does not change much during At, so the new angle at time t₁ is θι = θα + Ω, ΔΙ t₁ = ti-1 + At Ω and the angular velocity is the N₁1 — — sin(0₁-1)At L 0; = 0;−1 + Ω; ΔΙ i-1
Exercise 1: Complete the Euler-Cromer loop code below Write the code for calculating t¡,¡ and 0₁, based on the equations above. As before, we use a for loop for the repetitive (iterative) calculations. Remember that only indented lines become part of the loop, and all those lines must have the same indentation. • The sin function comes from the numpy library, so call it as np.sin(). • For the time step, use the variable dt defined earlier. • The i-th elements of time, angle and angular velocity can be accessed as t[i], theta[i] and Omega [i], respectively. For element number i 1, replace i with i-1. In [ ]: # Initial conditions. Vector numbering in Python starts at 0. time [0] = 0 theta [0] = Omega [0] # initial time np.deg2rad (5.0) # initial angle, converted to radians. = 0 # initial angular velocity (pendulum starts at rest) # Main loop: Euler algorithm and evaluation of exact solutions for v and y # # Your code should look like this (without the # signs, of course): # for i in range (1, t_steps): # time [i] = # Omega [i] # theta[i] # YOUR CODE HERE raise NotImplementedError()
Transcribed Image Text:Exercise 1: Complete the Euler-Cromer loop code below Write the code for calculating t¡,¡ and 0₁, based on the equations above. As before, we use a for loop for the repetitive (iterative) calculations. Remember that only indented lines become part of the loop, and all those lines must have the same indentation. • The sin function comes from the numpy library, so call it as np.sin(). • For the time step, use the variable dt defined earlier. • The i-th elements of time, angle and angular velocity can be accessed as t[i], theta[i] and Omega [i], respectively. For element number i 1, replace i with i-1. In [ ]: # Initial conditions. Vector numbering in Python starts at 0. time [0] = 0 theta [0] = Omega [0] # initial time np.deg2rad (5.0) # initial angle, converted to radians. = 0 # initial angular velocity (pendulum starts at rest) # Main loop: Euler algorithm and evaluation of exact solutions for v and y # # Your code should look like this (without the # signs, of course): # for i in range (1, t_steps): # time [i] = # Omega [i] # theta[i] # YOUR CODE HERE raise NotImplementedError()
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