Final answer:
The angular speed of the blade in radians per second is 491.80 radians/sec.
The linear speed of the saw teeth as they contact the wood is 169.05 feet/sec.
Step-by-step explanation:
(a) To find the angular speed of the blade in radians per second, we first need to convert the given revolutions per minute (RPM) to revolutions per second (RPS). We know that there are 60 seconds in a minute, so to convert RPM to RPS, we divide the given value by 60.
Therefore, the angular speed in revolutions per second is
4700/60 = 78.33 RPS.
The conversion factor from revolutions to radians is 2π radians per revolution. So, to find the angular speed in radians per second, we multiply the angular speed in RPS by 2π.
Therefore, the angular speed in radians per second is
78.33 x 2π = 491.80 radians/sec (rounded to two decimal places).
(b) To find the linear speed of the saw teeth as they contact the wood, we need to find the circumference of the circular blade. The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle. In this case, the diameter of the blade is 8.25 inches, so the radius is half of that, which is 4.125 inches.
To convert inches to feet, we divide by 12.
Therefore, the radius in feet is
4.125/12 = 0.34375 feet.
The linear speed is equal to the angular speed multiplied by the radius.
So, the linear speed in feet per second is 491.80 x 0.34375 = 169.05 feet/sec (rounded to two decimal places).