Final answer:
The centripetal acceleration at different radii of a rotating rod can be compared using the formula a = (v^2)/r. By converting the given angular velocity to linear velocity using v = ωr, we can calculate the centripetal accelerations at various radii. For example, at a radius of 1.0 m, the centripetal acceleration is approximately 158.11 m/s^2.
Step-by-step explanation:
The centripetal acceleration of an object moving in a circular path is given by the equation: a = (v^2)/r, where 'a' is the centripetal acceleration, 'v' is the linear velocity, and 'r' is the radius of the circular path. In this case, we are given the angular velocity of the rotating rod, which is given in revolutions per second. To compare the centripetal accelerations at different radii, we can first convert the angular velocity to linear velocity, using the formula: v = ωr, where 'ω' is the angular velocity and 'r' is the radius of the circular path.
(a) For a radius of 1.0 m, the linear velocity is 2.0 rev/s * 2π rad/rev * 1.0 m = 4π m/s ≈ 12.57 m/s. Using the centripetal acceleration formula, the centripetal acceleration at this radius is a = (12.57 m/s)^2 / 1.0 m = 158.11 m/s^2.
(b) For a radius of 2.0 m, the linear velocity is 2.0 rev/s * 2π rad/rev * 2.0 m = 16π m/s ≈ 50.27 m/s. The centripetal acceleration at this radius is a = (50.27 m/s)^2 / 2.0 m = 630.42 m/s^2.
(c) For a radius of 4.0 m, the linear velocity is 2.0 rev/s * 2π rad/rev * 4.0 m = 32π m/s ≈ 100.53 m/s. The centripetal acceleration at this radius is a = (100.53 m/s)^2 / 4.0 m = 2526.27 m/s^2.
(d) For a radius of 6.0 m, the linear velocity is 2.0 rev/s * 2π rad/rev * 6.0 m = 36π m/s ≈ 113.10 m/s. The centripetal acceleration at this radius is a = (113.10 m/s)^2 / 6.0 m = 3206.06 m/s^2.
(e) For a radius of 8.0 m, the linear velocity is 2.0 rev/s * 2π rad/rev * 8.0 m = 64π m/s ≈ 201.06 m/s. The centripetal acceleration at this radius is a = (201.06 m/s)^2 / 8.0 m ≈ 5042.61 m/s^2.