Final answer:
To achieve a centripetal acceleration of 10 g at a radius of 15.0 m, the angular velocity needs to be approximately 2.55 rad/s.
Step-by-step explanation:
Finding the Angular Velocity for a Centripetal Acceleration of 10 g
Let's find the angular velocity (ω) that will produce a centripetal acceleration of 10 g at a radius of 15.0 m from the center of rotation. First, we must recognize that 1 g is equal to the acceleration due to gravity, which is approximately 9.81 m/s². Therefore, 10 g is 10 × 9.81 m/s².
The formula for centripetal acceleration (ac) is given by:
ac = ω² × r, where r is the radius.
By rearranging the formula to solve for ω, we get:
ω = √(ac / r)
Substituting the given values into the equation gives us:
ω = √((10 × 9.81 m/s²) / 15.0 m) = √(98.1 m/s² / 15 m).
After calculating, we get an angular velocity of approximately 2.55 rad/s.