Question

The motor of a table saw is rotating at 3450 ${\rm rev/min}$ . A pulley attached to the motor shaft drives asecond pulley of half the diameter by means of a V-belt. A circularsaw blade of diameter 0.208 ${\rm m}$ is mounted on the same rotating shaft as the secondpulley.

The operator is careless and the bladecatches and throws back a small piece of wood. This piece of woodmoves with linear speed equal to the tangential speed of the rim ofthe blade. What is this speed?

v =_______________________ m/s

Calculate the radial acceleration of pointson the outer edge of the blade to see why sawdust doesn't stick toits teeth.

$\alpha$ =______________________m/s²

Answer 1

Concepts and reason

The concepts used to solve this problem are angular speed and radial acceleration.

First, find the linear speed by using the relation between radius and angular velocity. Then radius is calculated by using the diameter.

Finally, find the radial acceleration by using the relation between angular speed and radius.

Fundamentals

The relation between the linear velocity and the angular velocity is as follows:

Here, is the tangential velocity, r is the radius of the path and is the angular velocity.

From the question, the diameter of the second pulley is half that of the first pulley.

Hence, the angular velocity of the second pulley is twice of the first pulley. Therefore, the linear speed of the wood becomes as follows:

The radius of the circular saw blade is as follows:

Here, d is the diameter and r is the radius.

The expression for the radial acceleration is as follows:

Here, is the radial acceleration.

(1)

The linear speed of the wood is as follows:

…… (1)

Rewrite the above equation (1) by substituting for.

Substitute for and for to find .

(2)

The expression for the radial acceleration is as follows:

…… (2)

Replace for and for in the equation (2).

Substitute for and for to find .

Ans: Part 1

Therefore, the linear speed of the wood is.