Explanation It is given that the two concentric cylinders are centered along z ^ axis with radius R ± ε . It is represented in figure 1. It is given that the puck is contained between the two cylinders and is moving freely without any friction at a fixed distance from the central vertical axis. There are two forces acting on the puck, the normal force acting in the direction of ρ ^ and the force of gravity in the direction of z ^ . Write the expression for the force acting on the puck in ρ ^ direction. F ρ = N (I) Here, F ρ is the net force acting on the puck in ρ ^ direction and N is the normal force. Write the equation for the force acting on the puck in ρ ^ direction. F ρ = m ( ρ ˙ − ρ ϕ ˙ 2 ) (II) Here, m is the mass of the puck. The puck is contained to the cylinder so that its ρ coordinate will always will be R . Write the expression for the ρ coordinate of the puck. ρ = R (III) Here, R is the radius of the cylinder. Differentiate the above equation with respect to time. ρ ˙ = 0 (IV) Put the above equation in equation (II). F ρ = m ( 0 − ρ ϕ ˙ 2 ) = − m ρ ϕ ˙ 2 (V) Equate equations (I) and (V). N = − m ρ ϕ ˙ 2 Write the expression for the force acting on the puck in ϕ ^ direction. F ϕ = 0 (VI) Here, F ϕ is the net force acting on the puck in ϕ ^ direction. Write the equation for the force acting on the puck in ϕ ^ direction. F ϕ = m ( ρ ϕ ¨ − 2 ρ ˙ ϕ ˙ ) Put equations (III) and (IV) in the above equation. F ϕ = m ( R ϕ ¨ − 0 ) = m R ϕ ¨ (VII) Equate equations (VI) and (VII)

5th Edition

ISBN: 9781891389221

Author: John R. Taylor

Publisher: University Science Books

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Question

Chapter 1, Problem 1.49P

To determine

The expression for the general motion of the puck by writing and solving Newton’s second law.

Expert Solution & Answer

Chapter 1 Solutions

Classical Mechanics

Ch. 1 - Prob. 1.1P Ch. 1 - Prob. 1.2P Ch. 1 - Prob. 1.3P Ch. 1 - Prob. 1.4P Ch. 1 - Prob. 1.5P Ch. 1 - Prob. 1.6P Ch. 1 - Prob. 1.7P Ch. 1 - Prob. 1.8P Ch. 1 - Prob. 1.9P Ch. 1 - Prob. 1.10P

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