A 7.25 inch circular saw blade rotates at 5200 revolutions per minute.
a) Find the angular speed of the saw blade in radians per minute.
b) Find the linear speed in feet per minute of one of the 24 cutting teeth as they contact the wood being cut.
To find the angular speed of the saw blade in radians per minute, you need to convert the revolutions per minute to radians per minute.
a) Conversion from revolutions to radians:
Since there are 2π radians in one revolution, you can find the angular speed in radians per minute by multiplying the number of revolutions per minute by 2π:
Angular speed (in radians per minute) = 5200 revolutions per minute * 2π radians per revolution
b) To find the linear speed in feet per minute of one of the cutting teeth as it contacts the wood, you need to utilize the formula for linear speed:
Linear speed = radius * angular speed
The radius of the circular saw blade is half of its diameter. Therefore, the radius would be 7.25 inches / 2 = 3.625 inches.
To convert the radius from inches to feet, divide by 12:
Radius (in feet) = 3.625 inches / 12
Now, you have the angular speed in radians per minute and the radius in feet. You can calculate the linear speed in feet per minute:
Linear speed (in feet per minute) = Radius (in feet) * Angular speed (in radians per minute)
To find the angular speed of the saw blade in radians per minute, we need to convert the revolutions per minute to radians per minute.
a) Conversion from revolutions to radians:
1 revolution = 2π radians
So, the angular speed in radians per minute can be calculated as:
Angular speed (in radians/minute) = (Revolutions/minute) * (2π radians/revolution)
Given that the saw blade rotates at 5200 revolutions per minute, the angular speed is:
Angular speed = 5200 * 2π = 10400π radians/minute
b) To find the linear speed in feet per minute of one of the 24 cutting teeth as they contact the wood being cut, we first need to find the circumference of the circular saw blade.
Circumference of the circular saw blade = π * diameter
Given that the blade has a diameter of 7.25 inches, the circumference is:
Circumference = π * 7.25 inches
Since there are 24 cutting teeth around the blade, the linear speed can be calculated as the product of the angular speed and the circumference divided by the number of teeth:
Linear speed (in inches/minute) = (Angular speed * Circumference) / Number of teeth
Linear speed (in inches/minute) = (10400π radians/minute * 7.25 inches) / 24
To convert inches to feet, we divide by 12:
Linear speed (in feet/minute) = ((10400π radians/minute * 7.25 inches) / 24) / 12
Simplifying the equation gives the linear speed in feet per minute.
this is just like the carousel problem.
w = 2pi*f
s = w*r