Final answer:
The magnitude of the tangential acceleration of a point on the tip of the blade at time t = 0.206 s is 2.1072 m/s², which is calculated using the formula at = rα with the given angular acceleration and the radius of the fan blade.
Step-by-step explanation:
To calculate the magnitude of the tangential acceleration of a point on the tip of a fan blade at a given time, you need to use the relationship between tangential acceleration (at) and angular acceleration (α) which is given by at = rα, where r is the radius of the circle formed by the rotating fan blades.
In this case, the initial angular velocity is not directly relevant because we're calculating the tangential acceleration at a specific time, not the tangential velocity. The angular acceleration is given as 0.919 rev/s², which we first need to convert to rad/s² (since 1 rev = 2π rad): α = 0.919 rev/s² x 2π rad/rev = 5.7732 rad/s². The radius r of the circle formed by the fan blades is half the diameter, so r = 0.730 m / 2 = 0.365 m. Therefore, the tangential acceleration is at = 0.365 m x 5.7732 rad/s² = 2.1072 m/s².
The magnitude of the tangential acceleration of a point on the tip of the blade at time t = 0.206 s is therefore 2.1072 m/s².