angular velocity physics question PLEASE HELP??

Question

Dart hits wheel. In terms of m, M, v0, and R (of wheel), find:

 

A) angular speed of wheel
B) ratios of initial and final kinetic energies

Answer 1

Assume that: m is the mass of the dart and M is the mass of the wheel, R is the radius of the wheel and v0 is the velocity of the dart just before it strikes and sticks into the edge of the wheel. Assume that the wheel starts at rest.  Further assume that the rotation  axis of the wheel is the z axis and the direction of flight of the dart (before the collision) was along a line in the xy plane described by the equation  y = R.  Then this problem can be solved by invoking conservation of angular momentum about the z axis.

 

The initial angular momentum is  L_i =  m v0 R

 

 The final angular momentum is  L_f =  I ω    where ω is the angular speed of the wheel after the collision (rad/s) and    I (the rotational inertia) is  given by

 

   I  =  ½ M R² +  m R²  =   (½ M + m) R²  

 

   The first term in I  is the rotational inertial of  disk ( modeling the wheel as a disk of mass M and radius R).

    The second term in I is the contribution from the dart sticking into the wheel at the edge of the wheel.

 

  Setting   L_i =  L_f   an equation for ω results.  It is:

 

    ω = ( v0 /R) [m/(½M + m) ]

 

   The initial kinetic energy is KE_i =  ½ m v0².

 

    The final kinetic energy is  KE_f =  ½ I ω² =   ½ m v0² [ m/(½ M + m) ]

 

So the ratio   KE_f / KE_i =       m/(½ M + m)   which is a value less then unity- which is expected because this is an inelastic collision.