A small particle of mass m is attached by an inextensible cord to fixed vertical

Question

A small particle of mass m is attached by an inextensible cord to fixed vertical cylindrical shaft of radius a. The particle is given a high initial velocity in the horizontal plane. Initially, the angular velocity of the cord is omega _o and the distance from the point of tangency of the cord and shaft to the particle is r_0

a) One of the following quantities is conserved as the cord swings from the initial position to angle theta, which is it?

1) Linear momentum of the mass

2) angular momentum of the mass about some point

3) Energy of the mass

Explain in words why the other two are not conserved (just give an brief argument)

b) Using the appropriate conservation principle from part (a) determine omega = theta dot when the cord is at angle theta (in terms of the parameters of the initial configuration: a, r_0, omega_0)

c) Draw a free body diagram of the mass, and solve for the tension in the cord T at angle theta

Explanation / Answer

Angular momentum will conserve :::

b)   Li = initial angular mometum   = m*ro*wo

L2 is angular mometum at angle theta

       r = length of string = ro -a*theta

    L2 = m*(ro-a*theta)W

from cosnservation angular momentum ::;

   L2 = L1

m*(ro-a*theta)W   = m*ro*wo

                           W = ro*wo/(ro-a*theta)

C)   T = tension in string

        T = centripetal force

            = mrw^2

            = m*(ro -a*theta)*[ro*wo/(ro-a*theta)]^2

             = m(ro*wo)^2/((ro-a*theta)

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