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Chapter 4: Problem 63

The circular blade on a saw rotates at 5000 revolutions per minute. (a) Find the angular speed of the blade in radians per minute. (b) The blade has a diameter of \(7 \frac{1}{4}\) inches. Find the linear speed of a blade tip.

Short Answer

Expert verified
The angular speed of the blade is approximately 31415.93 radians per minute. The linear speed of the blade tip is around 113097.34 inches per minute.

Step by step solution

01

Convert revolutions per minute into radians per minute

To find the angular speed in radian per minute, use the formula: Angular speed = Number of rotations * \(2\pi\) radian. This involves multiplying the given number of revolutions (5000) per minute by \(2\pi\) radians because one rotation equals \(2\pi\) radians.
02

Perform the computation

The calculation should look like: Angular speed = 5000 * \(2\pi\). Perform the multiplication to find the angular speed.
03

Calculate the radius of the blade

The diameter of the blade is given as \(7 \frac{1}{4}\) inches. As the radius is half the diameter, the radius of the blade will be half of \(7 \frac{1}{4}\) inches.
04

Compute the linear speed

The formula for linear velocity of a rotating object is: Linear speed = Radius of rotation * Angular speed. Multiply the radius of the blade from step 3 (in inches) by the previously calculated angular speed (in radians per minute) to find the linear speed in inches per minute.

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