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question number 6.2 in page 279:
A drum of mass M_A and radius a rotates freely with initial angular velocity w_A(0). A second drum with mass M_B and radius b>a is mounted on the same axis and is at rest, although it is free to rotate.
A thin layer of sand with mass M_s is distributed on the inner surface of the amaller drum. At t=0, small perforations in the inner drum are opened. The sand starts to fly out at a constant rate \lambda and sticks to the outer drum. find the subsequent angular velocities of the two drums w_A and w_B. ignore the transit time of the sand.
ok obviously, we have conserved angular momentum and conserved linear momentum (no external force).
i.e I_Aw_A+I_Bw_B=I_Aw_A(0)
and (M_A+M_s)v0=(M_A-\lambdat)(w_A)a+(M_B+\lambdat)(w_B)b
where v0=w_A(0)*a
but i don't get it from the final answer which is the answer clue:
if lambda*t=M_b and b=2a then w_B=w_A(0)/8.
am i missing something here?
btw, the moments of inertia here are of a uniform think hoop, right?
which MR^2.
A drum of mass M_A and radius a rotates freely with initial angular velocity w_A(0). A second drum with mass M_B and radius b>a is mounted on the same axis and is at rest, although it is free to rotate.
A thin layer of sand with mass M_s is distributed on the inner surface of the amaller drum. At t=0, small perforations in the inner drum are opened. The sand starts to fly out at a constant rate \lambda and sticks to the outer drum. find the subsequent angular velocities of the two drums w_A and w_B. ignore the transit time of the sand.
ok obviously, we have conserved angular momentum and conserved linear momentum (no external force).
i.e I_Aw_A+I_Bw_B=I_Aw_A(0)
and (M_A+M_s)v0=(M_A-\lambdat)(w_A)a+(M_B+\lambdat)(w_B)b
where v0=w_A(0)*a
but i don't get it from the final answer which is the answer clue:
if lambda*t=M_b and b=2a then w_B=w_A(0)/8.
am i missing something here?
btw, the moments of inertia here are of a uniform think hoop, right?
which MR^2.