4. **Boomerang. If a boomerang is thrown correctly, it returns to the thrower. Most boomerangs consist...

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4. **Boomerang. If a boomerang is thrown correctly, it returns to the thrower. Most boomerangs consist of two arms separated by about 90 but the four-blade boomerang (see figure) has the same physics and is easier to analyze. To launch the boomerang, the thrower turns it sideways, grasps the forward blade, and throws it forward while snapping downward to give the boomerang spin. The figure shows a sideways view of the top of the boomerang shortly after it is launched, with V representing the forward center-of-mass velocity. (a) The velocity of each blade is determined by the sum of V and the angular rotation. Draw arrows beside the four blades that show the magnitude and direction of v = V+Vrotation. (b) Air that flows over the blade exerts a perpendicular "lift" force on each blade; see the second figure. (In this case, because the boomerang is sideways, "lift" is horizontal, not vertical.) The amount of lift increases with the speed of the air. Draw arrows on the figure that indicate the direction and relative size of the torque on each blade. (c) What direction is the net torque associated with "lift"? precession of the boomerang at the angular velocity wp. shows the direction of wp to the figure. This torque causes Add a vector that (d) Approximate each of the four blades of the boomerang as a stick of length 1. The total mass of the boomerang is M. What is the rotational inertia 3 about the central axis? (e) The net torque on the boomerang is approximately = cwV1, where c is a constant associated with the airfoil and w is the initial angular velocity. In terms of c, V, l, M, and w, what is wp? (f) When properly thrown, the boomerang completes a circle each time it pre- cesses through 2; in other words, wo = Wp, where wo is the frequency of the circular motion. Show that the radius of the circle is approximately R = == M 4cl 4. **Boomerang. If a boomerang is thrown correctly, it returns to the thrower. Most boomerangs consist of two arms separated by about 90 but the four-blade boomerang (see figure) has the same physics and is easier to analyze. To launch the boomerang, the thrower turns it sideways, grasps the forward blade, and throws it forward while snapping downward to give the boomerang spin. The figure shows a sideways view of the top of the boomerang shortly after it is launched, with V representing the forward center-of-mass velocity. (a) The velocity of each blade is determined by the sum of V and the angular rotation. Draw arrows beside the four blades that show the magnitude and direction of v = V+Vrotation. (b) Air that flows over the blade exerts a perpendicular "lift" force on each blade; see the second figure. (In this case, because the boomerang is sideways, "lift" is horizontal, not vertical.) The amount of lift increases with the speed of the air. Draw arrows on the figure that indicate the direction and relative size of the torque on each blade. (c) What direction is the net torque associated with "lift"? precession of the boomerang at the angular velocity wp. shows the direction of wp to the figure. This torque causes Add a vector that (d) Approximate each of the four blades of the boomerang as a stick of length 1. The total mass of the boomerang is M. What is the rotational inertia 3 about the central axis? (e) The net torque on the boomerang is approximately = cwV1, where c is a constant associated with the airfoil and w is the initial angular velocity. In terms of c, V, l, M, and w, what is wp? (f) When properly thrown, the boomerang completes a circle each time it pre- cesses through 2; in other words, wo = Wp, where wo is the frequency of the circular motion. Show that the radius of the circle is approximately R = == M 4cl