Sarah M.

asked • 05/22/15# angular velocity physics question PLEASE HELP??

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

B) ratios of initial and final kinetic energies

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## 1 Expert Answer

Richard P. answered • 05/22/15

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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 R

^{2}+ m R^{2}= (½ M + m) R^{2} 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 v0^{2}. The final kinetic energy is KE

_{f}= ½ I ω^{2}= ½ m v0^{2}[ 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.## Still looking for help? Get the right answer, fast.

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Mark K.

05/22/15