Final answer:
To find the turntable's angular velocity when the block reaches the outer edge, we can use the principle of conservation of angular momentum. By setting the initial and final angular momenta of the system equal to each other, we can solve for the final angular velocity. The specific equation to use depends on the masses and radii of the turntable and block.
Step-by-step explanation:
To find the turntable's angular velocity when the block reaches the outer edge, we can use the principle of conservation of angular momentum. The initial angular momentum of the system, which includes both the turntable and the block, is equal to the final angular momentum when the block reaches the outer edge. Since the turntable is initially rotating without the block, its initial angular momentum is given by:
Li = Iturntable * ωi
where Li is the initial angular momentum, Iturntable is the moment of inertia of the turntable, and ωi is the initial angular velocity of the turntable. Since the block is at the center, its initial angular momentum is zero.
When the block reaches the outer edge, its final angular momentum is given by:
Lf = Iblock * ωf
where Lf is the final angular momentum, Iblock is the moment of inertia of the block, and ωf is the final angular velocity of the turntable when the block reaches the outer edge.
Since angular momentum is conserved, we can set the initial and final angular momenta equal to each other:
Li = Lf
Iturntable * ωi = Iblock * ωf
Now we can solve for ωf:
ωf = (Iturntable * ωi) / Iblock
Substituting in the known values:
ωf = ((1/2) * Mturntable * Rturntable^2 * ωi) / (Mblock * Rblock^2)
where Mturntable is the mass of the turntable, Rturntable is the radius of the turntable, and Mblock and Rblock are the mass and radius of the block, respectively.
By substituting the known values into the equation, you can find the final angular velocity of the turntable when the block reaches the outer edge.