Explanation Given: The pendulum is released at θ = π 3 and the initial angular velocity is ω 0 radian per second. Calculation: Consider the angular displacement of the pendulum as θ ( t ) where, t is the time. Consider a pendulum of length l to which a mass m is attached at one end of the wire. From the figure 1 it is observed that as the mass ( m ) is released it moves under the influence of gravity and the angular acceleration is obtained as follows: a = d 2 s d t 2 = d 2 ( l θ ) d t 2 = l d 2 θ d t 2 So as per the Newton’s second law the force on the mass ( m ) is calculated as follows: F = m a = m l d 2 θ d t 2 The tangential component of the force due to the weight is as follows: W = − m g sin θ Equate the above two tangential forces as calculated above. m l d 2 θ d t 2 = − m g sin θ m l d 2 θ d t 2 + m g sin θ = 0 d 2 θ d t 2 + g l sin θ = 0 ( Divide by m l ) Label the above equation as follows: d 2 θ d t 2 + g l sin θ = 0 … … ( 1 ) Consider x = θ and y = θ ′ . From the above statement it is concluded that y = x ′ . Substitute x for θ and y for θ ′ in equation ( 1 ) . ( θ ′ ) ′ + g l sin x = 0 y ′ + g l sin x = 0 y ′ = − g l sin x d y d t = − g l sin x Use the chain rule to simplify the above expression. d y d x d x d t = − g l sin x y d y d x = − g l sin x ( x ′ = y ) Separate the variables in the above equation and integrate as follows: y d y = − g l sin x d x ∫ y d y = − g l ∫ sin x d x y 2 = − 2 g l ( − cos x ) + 2 c ′ y 2 = 2 g l cos x + c ( 2 c ′ = c ) Label the above equation as follows: y 2 = 2 g l cos x + c … … ( 3 ) It is given that initially the angular position is θ = π 3 and angular momentum is ω 0 that is y = ω 0

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

See similar textbooks

Videos

Question

Chapter 10.4, Problem 1E

To determine

The conditions in which the motion of the pendulum is periodic if the pendulum is released at $θ=π3$ and the initial angular velocity is $ω0$ radian per second.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.3, Problem 39E

Chapter 10.4, Problem 2E

Students have asked these similar questions

25.5. Find the Laurent series for the function 1/[(z - 1)(-2)(z - 3)] in the following domains: (a). 0 3. شهری

25.1. Expand each of the following functions f(z) in a Laurent series on the indicated domain: (a). z² - 2z+5 (2-2)(z² + 1)' (c). Log za 2 b (z - موجود 11, 29, where b>a> 1 are real, |z| > b.

25.3. Find the Laurent series for the function z/[(22 + 1)(z² + 4)] in the following domains (a). 02.

Chapter 10 Solutions

Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 10.1 - In Problems 16 write the given nonlinear...Ch. 10.1 - Prob. 2E Ch. 10.1 - In Problems 16 write the given nonlinear...Ch. 10.1 - Prob. 4E Ch. 10.1 - In Problems 16 write the given nonlinear...Ch. 10.1 - In Problems 16 write the given nonlinear...Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E

Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - Prob. 13E Ch. 10.1 - Prob. 14E Ch. 10.1 - Prob. 15E Ch. 10.1 - In Problems 716 find all critical points of the...Ch. 10.1 - In Problems 2326 solve the given nonlinear plane...Ch. 10.1 - In Problems 2326 solve the given nonlinear plane...Ch. 10.1 - In Problems 2326 solve the given nonlinear plane...Ch. 10.1 - In Problems 2326 solve the given nonlinear plane...Ch. 10.1 - Prob. 27E Ch. 10.1 - Prob. 28E Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.2 - In Problems 916 classify the critical point (0, 0)...Ch. 10.2 - Prob. 10E Ch. 10.2 - Prob. 11E Ch. 10.2 - Prob. 12E Ch. 10.2 - In Problems 916 classify the critical point (0, 0)...Ch. 10.2 - In Problems 916 classify the critical point (0, 0)...Ch. 10.2 - In Problems 916 classify the critical point (0, 0)...Ch. 10.2 - Prob. 16E Ch. 10.2 - Prob. 17E Ch. 10.2 - Determine a condition on the real constant so...Ch. 10.2 - Prob. 19E Ch. 10.2 - Prob. 20E Ch. 10.2 - Prob. 21E Ch. 10.2 - Prob. 22E Ch. 10.2 - Prob. 23E Ch. 10.2 - In Problems 23-26 a nonhomogeneous linear system...Ch. 10.2 - Prob. 25E Ch. 10.2 - Prob. 26E Ch. 10.3 - Prob. 1E Ch. 10.3 - Prob. 2E Ch. 10.3 - Prob. 3E Ch. 10.3 - Prob. 4E Ch. 10.3 - Prob. 5E Ch. 10.3 - In Problems 310, without solving explicitly,...Ch. 10.3 - In Problems 310, without solving explicitly,...Ch. 10.3 - Prob. 8E Ch. 10.3 - Prob. 9E Ch. 10.3 - In Problems 310, without solving explicitly,...Ch. 10.3 - Prob. 11E Ch. 10.3 - In Problems 1120 classify (if possible) each...Ch. 10.3 - Prob. 13E Ch. 10.3 - Prob. 14E Ch. 10.3 - Prob. 15E Ch. 10.3 - Prob. 16E Ch. 10.3 - Prob. 17E Ch. 10.3 - Prob. 18E Ch. 10.3 - Prob. 19E Ch. 10.3 - Prob. 20E Ch. 10.3 - Prob. 21E Ch. 10.3 - Prob. 22E Ch. 10.3 - Prob. 23E Ch. 10.3 - Prob. 24E Ch. 10.3 - Prob. 25E Ch. 10.3 - Prob. 26E Ch. 10.3 - Prob. 27E Ch. 10.3 - Show that the dynamical system x = x + xy y = 1 y...Ch. 10.3 - Prob. 29E Ch. 10.3 - Prob. 30E Ch. 10.3 - Prob. 31E Ch. 10.3 - Prob. 32E Ch. 10.3 - Prob. 33E Ch. 10.3 - Prob. 34E Ch. 10.3 - Prob. 35E Ch. 10.3 - Prob. 36E Ch. 10.3 - When a nonlinear capacitor is present in an...Ch. 10.3 - Prob. 38E Ch. 10.3 - Prob. 39E Ch. 10.4 - Prob. 1E Ch. 10.4 - Prob. 2E Ch. 10.4 - Prob. 3E Ch. 10.4 - Prob. 4E Ch. 10.4 - Prob. 5E Ch. 10.4 - Prob. 6E Ch. 10.4 - Prob. 7E Ch. 10.4 - Prob. 8E Ch. 10.4 - Prob. 9E Ch. 10.4 - Competition Models A competitive interaction is...Ch. 10.4 - Prob. 12E Ch. 10.4 - Prob. 13E Ch. 10.4 - Prob. 14E Ch. 10.4 - Prob. 15E Ch. 10.4 - Additional Mathematical Models Damped Pendulum If...Ch. 10.4 - Prob. 17E Ch. 10.4 - Prob. 18E Ch. 10.4 - Prob. 19E Ch. 10.4 - Prob. 20E Ch. 10.4 - Prob. 21E Ch. 10.4 - Prob. 22E Ch. 10 - Prob. 1RE Ch. 10 - Prob. 2RE Ch. 10 - Prob. 3RE Ch. 10 - Prob. 4RE Ch. 10 - Prob. 5RE Ch. 10 - Prob. 6RE Ch. 10 - Prob. 7RE Ch. 10 - Prob. 8RE Ch. 10 - Prob. 9RE Ch. 10 - Prob. 10RE Ch. 10 - Prob. 11RE Ch. 10 - Discuss the geometric nature of the solutions to...Ch. 10 - Classify the critical point (0, 0) of the given...Ch. 10 - Prob. 14RE Ch. 10 - Prob. 15RE Ch. 10 - Prob. 16RE Ch. 10 - Prob. 17RE Ch. 10 - Prob. 18RE Ch. 10 - Prob. 19RE Ch. 10 - Prob. 20RE

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The conditions in which the motion of the pendulum is periodic if the pendulum is released at θ = π 3 and the initial angular velocity is ω 0 radian per second.

Solution Summary: The author states that the motion of the pendulum is periodic if it is released at theta =pi

Videos

The conditions in which the motion of the pendulum is periodic if the pendulum is released at $θ=π3$ and the initial angular velocity is $ω0$ radian per second.

Want to see the full answer?

Chapter 10 Solutions