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Classical Mechanics | 0th Edition | ISBN: 9781891389221 | Authors: John R Taylor
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Consider the bead of Figure 4.13 threaded on a curved

Chapter 4, Problem 4.32

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QUESTION:

Consider the bead of Figure 4.13 threaded on a curved rigid wire. The bead's position is specified by its distance s, measured along the wire from the origin. (a) Prove that the bead's speed v is just v (Write v in terms of its components, dx/dt, etc., and find its magnitude using Pythagoras's theorem.) (b) Prove that m's' = Ftang, the tangential component of the net force on the bead. (One way to do this is to take the time derivative of the equation v2 = v v. The left side should lead you to :5' and the right to Ftang.) (c) One force on the bead is the normal force N of the wire (which constrains the bead to stay on the wire). If we assume that all other forces (gravity, etc.) are conservative, then their resultant can be derived from a potential energy U. Prove that Ftang = dU Ids. This shows that one- dimensional systems of this type can be treated just like linear systems, with x replaced by s and Fx by Ftang

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QUESTION:

Consider the bead of Figure 4.13 threaded on a curved rigid wire. The bead's position is specified by its distance s, measured along the wire from the origin. (a) Prove that the bead's speed v is just v (Write v in terms of its components, dx/dt, etc., and find its magnitude using Pythagoras's theorem.) (b) Prove that m's' = Ftang, the tangential component of the net force on the bead. (One way to do this is to take the time derivative of the equation v2 = v v. The left side should lead you to :5' and the right to Ftang.) (c) One force on the bead is the normal force N of the wire (which constrains the bead to stay on the wire). If we assume that all other forces (gravity, etc.) are conservative, then their resultant can be derived from a potential energy U. Prove that Ftang = dU Ids. This shows that one- dimensional systems of this type can be treated just like linear systems, with x replaced by s and Fx by Ftang

ANSWER:

Step 1 of 5

(a)

The position bead is specified by distance .

The smaller displacement  of beads in three dimensional coordinates is as follows.

                                                    

The velocity of the bead is given by,

                                                                    

The magnitude of the velocity is given by,

                                                                 

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