Question 452372: Two pulleys are connected by a belt. The radii of the pulleys asre 3 cm and 15 cm, and the distance between their centers is 24cm. Find the total length of belt needed to connect the pulleys.
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Two pulleys are connected by a belt. The radii of the pulleys are 3 cm and 15 cm, and the distance between their centers is 24cm. Find the total length of belt needed to connect the pulleys.
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The key here is to note that the point at which the belt just leaves or enters the pulleys, is a point of tangency, that is, the line connecting these points on the two pulleys are at right angles to the radii of both pulleys. Call this line x for my following calculations. Draw a line starting from the center of the 3 cm pulley parallel to the tangent line until it intersects with the 15 cm radius of the larger pulley. You now have a right triangle with legs x and 12, with hypotenuse the given distance between the centers of the pulleys=24. You now can calculate x with the pythagorean theorem or a trig function.
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All you need to do next is calculate the rest of the belt that remains on the pulleys.
By using the x-12-24 right triangle, you can determine the central angles of the arcs of each pulley that the belts are on. Note for the small pulley the central angle is <180º, and >180º for the larger pulley. That part of the belt remaining on the pulleys can now be calculated by dividing the central angle by 360º times the circumference of the respective pulleys. Calculations follow:
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x^2=24^2-12^2=576-144=432
x=√432=20.78 cm
Working with 15 cm pulley:
cosA=12/24=1/2
A=60º
Central angle of arc=360-2*60=360-120=240º
Circumference=2π15=94.23
Part of belt on larger pulley=(240/360)*94.3=62.87 cm
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Working with 3 cm pulley:
B=90-A=30º
Central angle=360-2*90-2*30=120º
Circumference=2π3=18.85
Part of belt on smaller pulley=(120/360)*18.85=6.28 cm
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ans:
Total length of belt=2*20.78+62.87+6.28=110.71 cm
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