In Problems 5 through 8, assume that the differential equation of a simple pendulum of length L is L0" + g0 = 0, where g = GM/R2 is the gravitational acceleration at the location of the pendulum (at distance R from the center of the earth; M denotes the mass of the earth). 5. Two pendulums are of lengths L1 and L2 and–when lo- cated at the respective distances R1 and R2 from the cen- ter of the earth-have periods p1 and p2. Show that P1 R1LI P2 R2/L2
In Problems 5 through 8, assume that the differential equation of a simple pendulum of length L is L0" + g0 = 0, where g = GM/R2 is the gravitational acceleration at the location of the pendulum (at distance R from the center of the earth; M denotes the mass of the earth). 5. Two pendulums are of lengths L1 and L2 and–when lo- cated at the respective distances R1 and R2 from the cen- ter of the earth-have periods p1 and p2. Show that P1 R1LI P2 R2/L2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Differential Equations
![In Problems 5 through 8, assume that the differential equation of a simple pendulum of length \( L \) is \( L\theta'' + g\theta = 0 \), where \( g = GM/R^2 \) is the gravitational acceleration at the location of the pendulum (at distance \( R \) from the center of the earth; \( M \) denotes the mass of the earth).
5. Two pendulums are of lengths \( L_1 \) and \( L_2 \) and—when located at the respective distances \( R_1 \) and \( R_2 \) from the center of the earth—have periods \( p_1 \) and \( p_2 \). Show that
\[
\frac{p_1}{p_2} = \frac{R_1 \sqrt{L_1}}{R_2 \sqrt{L_2}}.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2bafa3d4-8362-4eaf-b0f1-0c492e72839c%2F133ab4ef-7fe7-4bc8-a5e2-b0c6f4ad1e79%2Fn6e8ui3_processed.png&w=3840&q=75)
Transcribed Image Text:In Problems 5 through 8, assume that the differential equation of a simple pendulum of length \( L \) is \( L\theta'' + g\theta = 0 \), where \( g = GM/R^2 \) is the gravitational acceleration at the location of the pendulum (at distance \( R \) from the center of the earth; \( M \) denotes the mass of the earth).
5. Two pendulums are of lengths \( L_1 \) and \( L_2 \) and—when located at the respective distances \( R_1 \) and \( R_2 \) from the center of the earth—have periods \( p_1 \) and \( p_2 \). Show that
\[
\frac{p_1}{p_2} = \frac{R_1 \sqrt{L_1}}{R_2 \sqrt{L_2}}.
\]
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