A projectile of mass mp is traveling at a constant velocity vvec0 toward a stati

Question

A projectile of mass mp is traveling at a constant velocity vvec0 toward a stationary disk of mass M and radius R that is free to rotate its axis. Before impact, the projectile is traveling along a line displaced a distance b below the axis. The projectile strikes the disk and sticks to point B. Model the projectile as a point mass. (Use any variable or symbol stated above along with the following as necessary: v0 for the initial speed of the projectile.)

http://www.webassign.net/tipler6/10-p-066.gif (picture)

(a) Before impact, what is the total angular momentum of the disk-projectile system about the axis?
L0 =


(b) What is the angular speed ? of the disk-projectile system just after the impact?
? =

(c) What is the kinetic energy of the disk-projectile system after impact?
Kf =


(d) How much mechanical energy is lost in this collision?
?E =

Explanation / Answer

(a) Total angular momentum of the disk projectile system about the axis is              L = mp*vo*b (b) Consider the disk and projectile as a system, there is no external torque      acting on the system, angular momentum will conserve.           total inital angular momentum = total final angular momentum                   mp*vo*b = Itotal*                   mp*vo*b = [ (1/2)MR2 + mpb2 ]*                               = mp*vo*b / [ (1/2)MR2 + mpb2 ] (c) The kinetic energy of system after collision              Kf = (1/2)Itotal2                   = (1/2)[ (1/2)MR2 + mpb2 ] *(mp*vo*b )2/ [ (1/2)MR2 + mpb2 ]2                   = (mp*vo*b )2/ [ MR2 + 2mpb2 ] (d) Initial kinetic enrgy is Ki = (1/2)mp*vo2      Change in mechanical energy is          K = Kf - Ki                = (mp*vo*b )2/ [ MR2 + 2mpb2 ] - (1/2)mp*vo2                                                                         = (1/2)[ (1/2)MR2 + mpb2 ] *(mp*vo*b )2/ [ (1/2)MR2 + mpb2 ]2                   = (mp*vo*b )2/ [ MR2 + 2mpb2 ] (d) Initial kinetic enrgy is Ki = (1/2)mp*vo2      Change in mechanical energy is          K = Kf - Ki                = (mp*vo*b )2/ [ MR2 + 2mpb2 ] - (1/2)mp*vo2                        Change in mechanical energy is          K = Kf - Ki                = (mp*vo*b )2/ [ MR2 + 2mpb2 ] - (1/2)mp*vo2                                                                                  

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