Travel Rod
March 16th, 2010
PHYSICS! To what maximum angle, measured from vertical, does the rod (with the attached ball of clay) rotate?
A 78 g, 38-cm-long rod hangs vertically on a frictionless, horizontal axle passing through its center. A 11 g ball of clay traveling horizontally at 2.1 m/s hits and sticks to the very bottom tip of the rod.
To what maximum angle, measured from vertical, does the rod (with the attached ball of clay) rotate?
The clay ball has momentum 0.011*2.1 = 0.0231 kg-m/s just before it hits the tip of the rod. Its angular momentum with respect to the pivot point of the rod is 0.0231*0.38 = 8.78*10^-3 kg-m²/s.
The angular momentum of the stick and rod after collision must equal the initial angular momentum of the ball. The angular momentum of a rotating system is I*ω, where I is the moment of inertia of the system. The moment of inertia of the rod (mass Mr) plus ball (mass Mb) is
I = Mr*L²/3 + Mb*L² =(Mr/3 + Mb)*L²
I = 5.34*10^-3 kg-m²
Solve for the angular speed ω by equating the angular momentums:
I*ω = 8.78*10^-3 kg-m²/s
ω = 8.78*10^-3/5.34*10^-3 = 1.64 rad/s
The initial rotational energy of the system is 0.5*I*ω² = 0.00718 J
The potential energy of the system when it stops is elevation change of ball + elevation change of center of gravity of rod:
Mb*L*(1 - cosθ) + Mr*(L/2)*(1 - cosθ)
(1 - cosθ)*(Mb*L + Mr*L/2) = 0.00718
cosθ = 1 - 0.00718/(Mb*L + Mr*L/2) = 0.622
θ = 51.5º
NOTE: You cannot simply equate the initial kinetic energy of the ball with the system potential energy, since this is an inelastic collision and KE is not conserved.
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