The analytical solution (t) = 00 sin(wt) is based on the approximation sin≈ 0, which is valid with good accuracy for initial angles 00 ≤ 5°. Let's explore what happens when we start from large initial angles. • • First, make a copy of the for loop iteration and the plot code into the code cell below. (The code for initializing the arrays is already given.) Verify that your code runs. • • Next, change the initial angle theta [0] from 5° to larger values: first to 20°, then to 40, 60 and 80°. What do you observe? Keep in mind that the analytical solution is still based on the sin≈ 0 approximation. ■ Read the oscillation period from each graph. (Read the time for 5 or 10 oscillations, then divide that time by the number of oscillations.) ■ The Markdown cell below contains a table. Fill in the columns labeled "Period from analytical method" and "Period from numerical integration". Next, add your large angle experimental values (from your data sheet) to the table. (If you forgot to take these measurements in the lab, you can rig a simple pendulum at home, using string and some iron/steel/brass object, such as a steel bolt. Most smartphones and many wristwatches have built-in stopwatches.) • Which of your computed periods (analytical vs. numerical) are closest to the experimental values? Discuss your observations and conclusions. [ ]: # Initial conditions. Vector numbering in Python starts at 0. time [0] = 0 # initial time theta [0] = np.deg2rad(80.0) # initial angle, converted to radians. Omega [0] L = 0.4 = 0 # initial angular velocity (pendulum starts at rest) # pendulum length (m) theta_analyt = theta [0] np.cos (np.sqrt(g/L) *time) # Write your code below # YOUR CODE HERE raise NotImplementedError() # analytical solution Initial angle [°] Period from analytical method (s) Period from numerical integration (s) Experimental period (s) 20 40 52288 60 80 YOUR ANSWER HERE
The analytical solution (t) = 00 sin(wt) is based on the approximation sin≈ 0, which is valid with good accuracy for initial angles 00 ≤ 5°. Let's explore what happens when we start from large initial angles. • • First, make a copy of the for loop iteration and the plot code into the code cell below. (The code for initializing the arrays is already given.) Verify that your code runs. • • Next, change the initial angle theta [0] from 5° to larger values: first to 20°, then to 40, 60 and 80°. What do you observe? Keep in mind that the analytical solution is still based on the sin≈ 0 approximation. ■ Read the oscillation period from each graph. (Read the time for 5 or 10 oscillations, then divide that time by the number of oscillations.) ■ The Markdown cell below contains a table. Fill in the columns labeled "Period from analytical method" and "Period from numerical integration". Next, add your large angle experimental values (from your data sheet) to the table. (If you forgot to take these measurements in the lab, you can rig a simple pendulum at home, using string and some iron/steel/brass object, such as a steel bolt. Most smartphones and many wristwatches have built-in stopwatches.) • Which of your computed periods (analytical vs. numerical) are closest to the experimental values? Discuss your observations and conclusions. [ ]: # Initial conditions. Vector numbering in Python starts at 0. time [0] = 0 # initial time theta [0] = np.deg2rad(80.0) # initial angle, converted to radians. Omega [0] L = 0.4 = 0 # initial angular velocity (pendulum starts at rest) # pendulum length (m) theta_analyt = theta [0] np.cos (np.sqrt(g/L) *time) # Write your code below # YOUR CODE HERE raise NotImplementedError() # analytical solution Initial angle [°] Period from analytical method (s) Period from numerical integration (s) Experimental period (s) 20 40 52288 60 80 YOUR ANSWER HERE
Chapter2: The Kinetic Theory Of Gases
Section: Chapter Questions
Problem 98CP: Verify that v=8kBTm. Make the same scaling transformation as in the preceding problem.`
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![The analytical solution (t) = 00 sin(wt) is based on the approximation sin≈ 0, which is valid with good accuracy for initial angles 00 ≤ 5°. Let's explore what
happens when we start from large initial angles.
•
•
First, make a copy of the for loop iteration and the plot code into the code cell below. (The code for initializing the arrays is already given.)
Verify that your code runs.
•
•
Next, change the initial angle theta [0] from 5° to larger values: first to 20°, then to 40, 60 and 80°. What do you observe? Keep in mind that the analytical
solution is still based on the sin≈ 0 approximation.
■ Read the oscillation period from each graph. (Read the time for 5 or 10 oscillations, then divide that time by the number of oscillations.)
■ The Markdown cell below contains a table. Fill in the columns labeled "Period from analytical method" and "Period from numerical integration".
Next, add your large angle experimental values (from your data sheet) to the table. (If you forgot to take these measurements in the lab, you can rig a simple
pendulum at home, using string and some iron/steel/brass object, such as a steel bolt. Most smartphones and many wristwatches have built-in stopwatches.)
• Which of your computed periods (analytical vs. numerical) are closest to the experimental values? Discuss your observations and conclusions.
[ ]: # Initial conditions. Vector numbering in Python starts at 0.
time [0] = 0
# initial time
theta [0] = np.deg2rad(80.0) # initial angle, converted to radians.
Omega [0]
L = 0.4
= 0
# initial angular velocity (pendulum starts at rest)
# pendulum length (m)
theta_analyt = theta [0] np.cos (np.sqrt(g/L) *time)
# Write your code below
# YOUR CODE HERE
raise NotImplementedError()
# analytical solution
Initial angle [°] Period from analytical method (s) Period from numerical integration (s) Experimental period (s)
20
40
52288
60
80
YOUR ANSWER HERE](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc303f822-5e20-4d30-8920-178170554c96%2Fd9126395-9124-42b5-92eb-12014b3ca486%2F46toi4o_processed.png&w=3840&q=75)
Transcribed Image Text:The analytical solution (t) = 00 sin(wt) is based on the approximation sin≈ 0, which is valid with good accuracy for initial angles 00 ≤ 5°. Let's explore what
happens when we start from large initial angles.
•
•
First, make a copy of the for loop iteration and the plot code into the code cell below. (The code for initializing the arrays is already given.)
Verify that your code runs.
•
•
Next, change the initial angle theta [0] from 5° to larger values: first to 20°, then to 40, 60 and 80°. What do you observe? Keep in mind that the analytical
solution is still based on the sin≈ 0 approximation.
■ Read the oscillation period from each graph. (Read the time for 5 or 10 oscillations, then divide that time by the number of oscillations.)
■ The Markdown cell below contains a table. Fill in the columns labeled "Period from analytical method" and "Period from numerical integration".
Next, add your large angle experimental values (from your data sheet) to the table. (If you forgot to take these measurements in the lab, you can rig a simple
pendulum at home, using string and some iron/steel/brass object, such as a steel bolt. Most smartphones and many wristwatches have built-in stopwatches.)
• Which of your computed periods (analytical vs. numerical) are closest to the experimental values? Discuss your observations and conclusions.
[ ]: # Initial conditions. Vector numbering in Python starts at 0.
time [0] = 0
# initial time
theta [0] = np.deg2rad(80.0) # initial angle, converted to radians.
Omega [0]
L = 0.4
= 0
# initial angular velocity (pendulum starts at rest)
# pendulum length (m)
theta_analyt = theta [0] np.cos (np.sqrt(g/L) *time)
# Write your code below
# YOUR CODE HERE
raise NotImplementedError()
# analytical solution
Initial angle [°] Period from analytical method (s) Period from numerical integration (s) Experimental period (s)
20
40
52288
60
80
YOUR ANSWER HERE
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