# Solving Crossed Belt Connecting Pulleys & Saw Arbor Revs/Min

• Marioqwe
The speed of the motor is given. So the speed of the saw is the speed of the motor times the ratio of the two pulley radii (saw over motor) times a negative sign because the directions of rotation are opposite. Or if you want to be more mathematical you could say the speed of the saw is -1 times the speed of the motor times the ratio of the radii of the saw and motor. The speed of the saw is a negative number because as the motor turns clockwise the saw turns counter-clockwise. So the speed of the saw is negative if the speeds of the motor and saw are in the same direction and positive if they are in opposite directions. Speeds of 0 mean
Marioqwe
A crossed belt connects a 20-cm pulley (10-cm radius) on an electric motor with a 40-cm pulley (20-cm radius) on a saw arbor. The electric motor runs at 1700 revolutions per minute.

(a) Determine the number of revolutions per minute of the saw.
(b) How does crossing the belt affect the saw in relation to the motor?
(c) Let L be the total length of the belt. Write L as a function of ?, where ? is measured in radians.

## The Attempt at a Solution

a) 3400 revs/min
b) it increases the angular velocity of the saw?
c)
I used the formula for circumference and arc length to find the length of the belt around both pulleys, but I don't know how to get the length of the belt that is not around the pulleys in terms of phi.

I don't know if I'm asking for too much but please, don't give the answer, just a hint.

Thank You

a) No I believe it would be half that, not double. Say the motor rotates once, with a radius r this means the pulley has moved a distance of $2\pi r$ since the circumference is just that. The saw has a circumference of $2\pi (2r)=2(2\pi r)$ so it will only rotate half a revolution since its circumference is double that of the motor.

c) Let's say the length of the belt from one pulley to the other is x, and we cut this length up into two parts, x1 and x2. The cut is made where the belts cross.

$$x=x_1+x_2$$

Now looking at the belt closest to the motor, let this pulley have a radius r. Now we already know (and it shows it on the diagram) that the belt creates a tangent to the pulley, which is perpendicular to the line running through the centre and through the point where the belt leaves the pulley. So we are dealing with a right triangle, where one angle is $\phi$.

Using trig, $$tan\phi=\frac{r}{x_1}$$ so $$x_1=r cot\phi$$

Now do the same for the saw, and then add those lengths together to get the entire length of party of the belt from one pulley to the other. And because of symmetry, the other part of the belt crossing is the same length.

For part b, look at the directions of the motor and saw. Then, look at the directions if the belt were not crossed.

How would you find the length of the belt around the pulley?

cedar2 said:
How would you find the length of the belt around the pulley?

Notice that the point where the belt leaves the pulley, it create a right angle to the centre of the pulley (a line tangent to a circle is perpendicular to its radius at point of intersection) and since you know that one of the angles is $\phi$, you'll realize that you're dealing with a right-triangle. You can do this for every part of the belt not touching the pulleys.

As for the part of the belt touching the pulleys, you should be able to find the angle subtended by the two radii in each pulley, so then you should be able to find the length in terms of that.

The speed of the saw depends only upon the speed of the motor and the ratio of the radii of the two pulleys. The fact the belt is crossed changes the direction of the turn- if the motor pulley is turning clockwise the saw pulley is turning counter-clockwise and vice-versa.

## 1. What is the purpose of solving for crossed belt connecting pulleys and saw arbor revs/min?

The purpose of solving for crossed belt connecting pulleys and saw arbor revs/min is to determine the optimal speed and configuration for a pulley and saw system. This can help ensure efficient and safe operation of the system.

## 2. What factors should be considered when solving for crossed belt connecting pulleys and saw arbor revs/min?

Some factors that should be considered when solving for crossed belt connecting pulleys and saw arbor revs/min include the size and speed of the pulleys, the type of belt being used, the desired speed of the saw, and the tension of the belt.

## 3. How do you calculate the speed of the saw arbor in a crossed belt pulley system?

The speed of the saw arbor in a crossed belt pulley system can be calculated by using the formula: Saw Arbor RPM = Motor RPM x Motor Pulley Diameter / Saw Arbor Pulley Diameter.

## 4. Can the number of pulleys in a crossed belt system affect the speed of the saw arbor?

Yes, the number of pulleys in a crossed belt system can affect the speed of the saw arbor. The more pulleys involved, the more friction and energy loss there will be, resulting in a slower speed of the saw arbor.

## 5. What are some potential issues that can arise when solving for crossed belt connecting pulleys and saw arbor revs/min?

Some potential issues that may arise when solving for crossed belt connecting pulleys and saw arbor revs/min include inaccurate calculations due to incorrect measurements or assumptions, incorrect belt tension leading to slippage, and inadequate power to drive the system resulting in slower than desired saw speed.

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