A circular power saw has an 8-1/2

inch diameter blade that rotates at 4400 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.

I'd appreciate any help on this problem. I have 5 more like it, so I really just want to know how to go about it so I can do the rest. Thanks!

Well, well, well. It seems like you're in luck today, my friend. As Clown Bot, I can surely help you solve this problem in a jolly way! Let's dive in, shall we?

a) Angular speed is given by the formula: angular speed = (2π * revs) / time. Since the problem already states that the blade rotates at 4400 revs per minute, we can substitute that value into our formula: angular speed = (2π * 4400) / 1 minute. Now, simplify that expression and you'll have your angular speed in radians per minute.

b) To find the linear speed, we need to determine the distance traveled by one tooth in one revolution. The formula for circumference is C = 2πr, where r is the radius of the circular saw's blade (half of the diameter). Substituting 8-1/2 inches (which is equal to 8.5/12 feet) for the diameter, we get r = (8.5/12) / 2 feet. Then, multiply the circumference by the number of teeth to find the total distance traveled by the teeth in one revolution. Finally, divide that total distance by the time for one revolution to find the linear speed in feet per minute.

Remember, my dear compadre, practice makes perfect. Once you've mastered this problem, the rest will be a piece of cake. Good luck!

Sure, I can help you with this problem.

a) To find the angular speed of the saw blade in radians per minute, we need to convert the revolutions per minute (RPM) to radians per minute (rad/min).

There are 2π radians in one revolution, so we can use this conversion factor:

Angular speed (in rad/min) = RPM * (2π radians/1 revolution)

In this case, the RPM is given as 4400. Plugging in the values:

Angular speed = 4400 RPM * (2π radians/1 revolution)

Simplifying:

Angular speed = 4400 * 2π rad/min

Therefore, the angular speed of the saw blade is 8800π rad/min.

b) To find the linear speed in feet per minute of one of the cutting teeth as they contact the wood being cut, we need to find the circumference of the circular motion.

The circumference of a circle is given by the formula C = πd, where d is the diameter of the circle.

In this case, the diameter of the blade is given as 8-1/2 inches. We need to convert it to feet, as the linear speed is required in feet per minute.

8-1/2 inches = 8.5 inches

1 foot = 12 inches, so we can convert the diameter to feet:

Diameter (in feet) = 8.5 inches / 12

Now we can calculate the circumference:

Circumference = π * Diameter

Now, to find the linear speed, we need to multiply the circumference by the angular speed:

Linear speed = Circumference * Angular speed

Plugging in the values:

Linear speed = π * Diameter * Angular speed

Simplifying:

Linear speed = π * (8.5/12) * (8800π)

Therefore, the linear speed of one of the cutting teeth as they contact the wood being cut is π * (8.5/12) * (8800π) feet per minute.

I hope this helps! Let me know if you have any other questions.

To find the angular speed of the saw blade in radians per minute, you need to convert the revolutions per minute into radians per minute.

We know that one revolution is equivalent to 2π radians.

a) To find the angular speed in radians per minute:
Angular speed = (Revolutions per minute) * (2π radians/1 revolution)
Angular speed = (4400 revolutions/minute) * (2π radians/1 revolution)

Now, we can calculate the angular speed:

Angular speed = (4400 revolutions/minute) * (2π radians/1 revolution)
Angular speed = (4400 * 2π) radians/minute

b) To find the linear speed in feet per minute of one of the cutting teeth, we need to calculate the distance covered by a cutting tooth in one minute.
The circumference of the circular saw blade represents the distance covered by one revolution.

Circumference = π * diameter
Circumference = π * 8.5 inches

To convert inches to feet, we divide the circumference by 12:
Circumference = (π * 8.5 inches) / 12

Now, we can calculate the linear speed of one cutting tooth:

Linear speed = (Circumference of the blade) * (Revolutions per minute) / (Number of teeth)
Linear speed = [(π * 8.5 inches) / 12] * (4400 revolutions/minute) / 24

Note: Make sure to simplify the units and conversions to get the final result in the desired units (radians per minute for part a, and feet per minute for part b).

Once you have the formulas and the values, you can substitute the values into the equations and calculate the answers.

a) one rotation = 2π radians

so 4400 rpm = 2π(4400) radians/min
= 8800π rad/min
= appr 27646 rad/min

b) one rotation = 2π(8.5) inches
= 17π inches
so 4400 rotations would be 17π(4400) inches
= 74800π

so the linear speed of a tooth = 74800π inches/min