Question

A projectile of mass and an initial velocity of collides with the end of a blade attached to a turbine. The rotational inertia of the turbine is . Assume the loss of energy of the projectile in the collision is completely transferred to the blades, causing them to spin. If the final rebound velocity of the projectile after hitting a turbine blade is , which of the following is most nearly the rotational velocity of the turbine after the collision?

1) 3.2 radians per second
2) 5.5 radians per second
3) 10.0 radians per second
4) 15.0 radians per second

Answer 1

The rotational velocity of the turbine after the collision is approximately 5.5 radians per second.

Step-by-step explanation:

The rotational velocity of the turbine can be calculated using the principle of conservation of angular momentum. The initial angular momentum of the projectile is equal to the final angular momentum of the turbine. The initial angular momentum of the projectile can be calculated using the formula ℓi = Ip × ωi, where Ip is the rotational inertia of the projectile and ωi is the initial rotational velocity. The final angular momentum of the turbine can be calculated using the formula ℓf = It × ωf, where It is the rotational inertia of the turbine and ωf is the final rotational velocity. Since the loss of energy of the projectile is completely transferred to the blades causing them to spin, the initial kinetic energy of the projectile can be equated to the final rotational kinetic energy of the turbine. Using the formula KEf = (1/2)Itωf2 and substituting the values given in the question, we can solve for ωf. The most nearly rotational velocity of the turbine after the collision is approximately 5.5 radians per second (option 2).