Wheels of an engine is 13 inch in diameter, and the pulley is 5 inches in diameter2. If the shafts of the drive wheel and the pulley are 2 feet apart, what length of belt is required to join them

Question

Book answer is 76.94 inches, but I don't know how to get that answer.

Answer 1

Draw a picture:

(since the diameters are listed, I divided them by 2 to get a radius for each. Also, the distance between the two are 2 feet which I have shown as 24 inches to get everything in one unit of measure)

The lengths of curved ends of the belt are easy, just take half of the circumference for each circle (C = 2πr so half the circumference is π•r for each)

The straight sections of the belt form a right triangle where the hypotenuse is the length of the straight section of the belt. The base of the triangle is 24. The height of the triangle is 6.5 - 2.5 (or 4). Use the Pythagorean Theorem to find the hypotenuse. h² = 4² + 24² . Don't forget there are two straight sections, both the same length.

Answer 2

NOTE: this could just as easily been about the gears on a bicycle connected by a chain, same concept, different size circles and different distance between the circles.

The total length of belt required to connect the drive wheel and pulley is equal to half circumference of drive wheel plus half the circumference of pulley plus two times the length it takes to connect the two (which is described by the hypotenuse of the right triangle formed by the belt sitting on the edges of the two wheels).

r = d / 2

1. r_{drive wheel} = 13" / 2 = 6.5"
2. r_pulley = 5" / 2 = 2.5"

circumference of a circle = 2 π r = d π

1. C_{drive wheel} = 13" π
2. C_pulley = 5" π

To determine the length between, the right triangle can be described as follows:

1. the base of the right triangle is 2 feet, or 24 inches
2. the height of the right triangle is the difference of the wheels' radii, 6.5" - 2.5" = 4"
3. the hypotenuse of the right triangle can be determined with the Pythagorean Theorem, solve for c

_{______________}

hypotenuse = √ (24")² + (4")² ~= 24.33"

The total length of pulley required = 2(24.33") + 0.5(13π) + 0.5(5π) ~= 76.94"

^{The total length of belt required to connect the drive wheel and pulley is equal to half circumference of drive wheel plus half the circumference of pulley plus two times the length it takes to connect the two}