The next seven questions pertain to the situation described belovw A horizontal

Question

The next seven questions pertain to the situation described belovw A horizontal turntable is made from a uniform solid disk and is initially rotating with angular velocity of 8.7 rad/s about a fixed vertical axis through its center. The turntable has a radius of 0.3 m and a moment of inertia of 0.1035 kg m- about the rotation axis. A piece of clay, initially at rest, is dropped onto the turntable and sticks to it at a distance d- 0.23 m from its center as shown in the figure. The final angular velocity of the turntable and the clay after the collision is 4.2 rad/s. Treat the piece of clay as a point particle and assume that there are no external torques acting on the system around the rotation axis 1) Which of these quantities are conserved during the collision? a. The total linear momentum of the clay and the turntable b. The total angular momentum of the clay and the turntable about the rotation axis. C. The angular momentum of the turntable about the rotation axis. d. The angular momentum of the clay about the rotation axis e. The mechanical energy of the clay and the turntable 2) What is the mass of the turntable? 3) What is the angular momentum of the turntable about the rotation axis before the clay is dropped? kg m'/s 4) What is the total angular momentum of the turntable and the clay about the rotation axis after the collision? kg m'/s

Explanation / Answer

let
wi = 8.7 rad/s
I = 1.035 kg.m^2
R = 0.3 m
d = 0.23 m
wf = 4.2 rad/s

1) b) The total angular momentum of the clay and the turnable about the rotaion axis.

2) let M is the mass of the turnable.

I = (1/2)*M*R^2

==> M = 2*I/R^2

= 2*0.1035/0.3^2

= 2.3 kg

3) Initial angular momentum, Li = Ii*wi

= 0.1035*8.7

= 0.90 kg.m^2/s

4) final angular momentum = initial angular momentum

= 0.90 kg.m^2/s

5) Lf = Li

If*wf = Ii*wi

If = Ii*wi/wf

= 0.1035*8.7/4.2

= 0.214 kg.m^2

6) If = 0.1035 + m*d^2

0.214 = 0.1035 + m*0.23^2

==> m = (0.214 - 0.1035)/0.23^2

= 2.09 kg

7) a) at the outside edge of the turnable

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