Final answer:
The tip of the blade moves through a distance of 2πr in one revolution. The tip's speed is given by Speed = (2πnr) / 60 meters per second. The magnitude of the tip's acceleration is a = 4π^2n^2r meters per second-squared. The period of the motion is Period = (1/n) * 60 seconds.
Step-by-step explanation:
(a) The tip of the blade moves through a distance equal to the circumference of a circle with radius r in one revolution. The formula for the circumference of a circle is C = 2πr, where r is the radius of the circle. Therefore, the distance the tip of the blade moves in one revolution is 2πr.
(b) The tip's speed can be calculated by dividing the distance traveled in one revolution by the time taken for one revolution. Since the fan completes n revolutions every minute, the time taken for one revolution is 1/n minutes. So, the tip's speed is given by Speed = (2πr) / (1/n) = 2πnr meters per minute. To convert this to meters per second, we divide by 60, giving the final answer as Speed = (2πnr) / 60 meters per second.
(c) The magnitude of the tip's acceleration is given by the formula a = (v^2) / r, where v is the tip's speed. Substituting the expression for speed derived in part (b), we get a = ((2πnr)^2) / r = 4π^2n^2r meters per second-squared.
(d) The period of the motion is the time taken for one revolution. As mentioned in part (b), the time taken for one revolution is 1/n minutes. To convert this to seconds, we multiply by 60, giving the final answer as Period = (1/n) * 60 seconds.