A crossed belt connects a 24-centimeter pulley (x = 12−cm radius) on an electric motor with a 48-centimeter pulley (y = 24-cm radius) on a saw arbor

Question

A crossed belt connects a 24-centimeter pulley (x = 12−cm radius)

 on an electric motor with a 48-centimeter pulley (y = 24-cm radius)

 on a saw arbor, as shown in the figure below. The electric motor runs at 2,100 revolutions per minute.

(a) Determine the number of revolutions per minute of the saw.

(b) Let L be the total length of the belt. Write L as a function of 𝜑, where 𝜑 is measured in radians. (Hint: Add the lengths of the straight sections of the belt and the length of the belt around each pulley.)

L = 

What is the domain of the function? (Enter your answer using interval notation.)

(c) Use a graphing utility to complete the table. (Round your answers to one decimal place.)

𝜑	0.3	0.6	0.9	1.2	1.5
L

(d) Use a graphing utility to graph the function over the appropriate domain.

(e) Find lim 𝜑→(𝜋/2)⁻ L.

 

Answer 1

Hi Anikka C.,

Actually, the limit of L as phi approaches .pi./2 is not physically realizable. The two pulleys will be touching, BUT the belt must cross through itself (on the same in-out position). Unless one of the pulleys is tilted up-down, perhaps? That will offset the belt positions at insertion onto the pulleys.

So given that mechanical difficulty, why might a manufacturer choose to use a crossed-belt arrangement? (There;s a good reason....)

-- Cheers, --Mr. d.