A belt is wrapped around a 6.25" pulley and has 220 degrees of contact. How much of the belt is contacting the pulley?

To determine how much of the belt is contacting the pulley, it's essential to calculate the length of the arc corresponding to the angle of contact on the pulley. The formula to find the length of the arc (L) for a circle is given by:

[ L = r \cdot \theta ]

where ( r ) is the radius and ( \theta ) is the angle in radians.

First, convert the diameter of the pulley into a radius. The diameter is 6.25 inches, so the radius will be half of that:

[ r = \frac{6.25}{2} = 3.125 \text{ inches} ]

Next, convert the contact angle from degrees to radians. Knowing that ( 180^\circ ) is equivalent to ( \pi ) radians, you can calculate:

[ \theta = \frac{220 \times \pi}{180} \approx 3.8397 \text{ radians} ]

Now plug the radius and the angle in radians into the arc length formula:

[ L = 3.125 \cdot 3.8397 \approx 12.0003 \text{ inches} ]

Rounding gives you approximately 12