Question 7.SP.34: Figure 7-31 shows a remote-controlled 5-kg object at a dista......
Figure 7-31 shows a remote-controlled 5-kg object at a distance of 1.5 m from the center of a horizontal “weightless” turntable that is turning 2 rad/s about a vertical axis. An opposing moment M = 3t N⋅m is applied to the shaft. Determine (a) how long it will take for the turntable to reach a speed of 1.5 rad/s and (b) how far the object must be moved, if the moment is removed, to bring the turntable back to a speed of 2 rad/s.
(a) The angular momentum of the object is the product of its linear momentum m v and its radial distance r . The angular impulse equals the change in angular momentum. Hence, using v=r \omega , we have
\int_{0}^{t}-3 t d t=5(1.5 \times 1.5)(1.5)-5(1.5 \times 2)(1.5) \quad \therefore t=\underline{1.94 \mathrm{~s}}
(b) To solve the second part of the problem, use conservation of angular momentum to determine the necessary radius r at which to place the object:
5(1.5)(1.5)(1.5)=5(1.5)(2) r \quad \therefore r=1.125 \mathrm{~m}
The object must be moved from 1.5 to 1.125 m, or \underline{0.375 m} toward the center.