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The period of a pendulum with length L that makes a maximum angle  θ_0 with the vertical is T  =  4√Lg ∫0π/2 dx/√1 − k^2 sin^2 x where  k  =  sin (1/2θ_0) and g is the acceleration due to gravity. Use the inequalities in part (b) to estimate the period of a pendulum with  L  =  1 meter and θ_0  =  10∘. How does it compare with the estimate  T  ≈  2π√L/g? What if θ_0  =  42∘? Part(b):  2π√L/g (1 + 1/4k^2)⩽  T  ⩽  2π√L/g4 − 3k^2/4 − 4k^2

The period of a pendulum with length L that makes a maximum angle  θ_0 with the vertical is T  =  4√Lg ∫0π/2 dx/√1 − k^2 sin^2 x where  k  =  sin (1/2θ_0) and g is the acceleration due to gravity. Use the inequalities in part (b) to estimate the period of a pendulum with  L  =  1 meter and θ_0  =  10∘. How does it compare with the estimate  T  ≈  2π√L/g? What if θ_0  =  42∘? Part(b):  2π√L/g (1 + 1/4k^2)⩽  T  ⩽  2π√L/g4 − 3k^2/4 − 4k^2

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.4: Values Of The Trigonometric Functions
Problem 40E
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The period of a pendulum with length L that makes a maximum angle

 θ_0 with the vertical is T  =  4√Lg ∫0π/2 dx/√1 − k^2 sin^2 x
where  k  =  sin (1/2θ_0) and g is the acceleration due to gravity. Use the inequalities in part (b) to estimate the period of a pendulum with
 L  =  1 meter and θ_0  =  10∘.
How does it compare with the estimate  T  ≈  2π√L/g? What if θ_0  =  42∘?

Part(b):  2π√L/g (1 + 1/4k^2)⩽  T  ⩽  2π√L/g4 − 3k^2/4 − 4k^2
(AS 32. The figure shows a pendulum with length L that makes a maximum angle 0, with the vertical. Using Newton's Second Law it can be shown that the period T (the time for one complete swing) is given by dx T= 4, a Jo VI - k² sin³x where k = sin( 0o) and g is the acceleration due to gravity. If L = 1 m and 0, = 42°, use Simpson's Rule with n = 10 to find the period. | 6,
Transcribed Image Text:(AS 32. The figure shows a pendulum with length L that makes a maximum angle 0, with the vertical. Using Newton's Second Law it can be shown that the period T (the time for one complete swing) is given by dx T= 4, a Jo VI - k² sin³x where k = sin( 0o) and g is the acceleration due to gravity. If L = 1 m and 0, = 42°, use Simpson's Rule with n = 10 to find the period. | 6,
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