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A simple pendulum consists of a light wire of length 1 at the end of which is suspended a ball of mass m making an angle with the vertical. The equations of motion for a simple pendulum along the wire and perpendicular to it are mg cos - T -mg sin where I is the tension in the wire and g is the gravitational constant. (i) Show that for small displacement and velocity the pendulum is governed by the equation of motion = = - -ml0², mlö, E j² ö+w²0 = 0, (4) where w = √√√g/l. Make sure to explain what happens to both of the equations of motion (2) and (3). (ii) Show that 62 (2) (3) 02 2 is a conserved quantity of the motion described by equation (4).

A simple pendulum consists of a light wire of length 1 at the end of which is suspended a ball of mass m making an angle with the vertical. The equations of motion for a simple pendulum along the wire and perpendicular to it are mg cos - T -mg sin where I is the tension in the wire and g is the gravitational constant. (i) Show that for small displacement and velocity the pendulum is governed by the equation of motion = = - -ml0², mlö, E j² ö+w²0 = 0, (4) where w = √√√g/l. Make sure to explain what happens to both of the equations of motion (2) and (3). (ii) Show that 62 (2) (3) 02 2 is a conserved quantity of the motion described by equation (4).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.5: Trigonometric Graphs
Problem 29E
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Question
A simple pendulum consists of a light wire of length 1 at the end of which is suspended a ball of mass m making an angle with the vertical. The equations of motion for a simple pendulum along the wire and perpendicular to it are mg cos - T -mg sin 0 where I is the tension in the wire and g is the gravitational constant. (i) Show that for small displacement and velocity the pendulum is governed by the equation of motion = = = E =-=-0² mii , Ö+w²0=0, (4) where w = √g/1. Make sure to explain what happens to both of the equations of motion (2) and (3). (ii) Show that + miö, 6² is a conserved quantity of the motion described by equation (4). 2
Transcribed Image Text:A simple pendulum consists of a light wire of length 1 at the end of which is suspended a ball of mass m making an angle with the vertical. The equations of motion for a simple pendulum along the wire and perpendicular to it are mg cos - T -mg sin 0 where I is the tension in the wire and g is the gravitational constant. (i) Show that for small displacement and velocity the pendulum is governed by the equation of motion = = = E =-=-0² mii , Ö+w²0=0, (4) where w = √g/1. Make sure to explain what happens to both of the equations of motion (2) and (3). (ii) Show that + miö, 6² is a conserved quantity of the motion described by equation (4). 2
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