A 4 m long shaft carries three pulleys, two at its ends and third at the midpoint. The two end pulleys have mass of 80 and 40 kg and their centre of gravity are 3 and 5 mm, respectively, from the axis of the shaft. The middle pulley mass is 50 kg and its center of gravity is 8 mm from the shaft axis

Question

A 4 m long shaft carries three pulleys, two at its ends and third at the midpoint. The two end pulleys have mass of 80 and 40 kg and their centre of gravity are 3 and 5 mm, respectively, from the axis of the shaft. The middle pulley mass is 50 kg and its center of gravity is 8 mm from the shaft axis. The pulleys are keyed to the shaft and the assembly is in static balance. The shaft rotates at 300 rpm in two bearings 2.5 m apart with equal overhang on either side. Determine (a) the relative angular positions of the pulleys and (b) dynamic reactions at the two bearings.

Accepted Answer

[latex] ext { For static balance, } M_{ E }=0 ext {, gives (Fig.12.39). } [/latex]

[latex] R_{D} imes 2.5=(80 imes 4.25+50 imes 1.25-40 imes 0.75) imes 9.81 [/latex].

[latex] R_{D}=1147.8 N , R_{E}=519.9 N [/latex].

[latex] \omega=\frac{2 \pi imes 300}{60}=31.416 rad / s [/latex].

[latex] F_{A}=80 imes 0.003 imes(31.416)^{2}=236.87 N [/latex].

[latex] F_{B}=50 imes 0.008 imes(31.416)^{2}=394.78 N [/latex].

[latex] F_{C}=40 imes 0.005 imes(31.416)^{2}=197.39 N [/latex].

[latex] R_{D} imes 2.5=236.87 imes 4.25+394.78 imes 1.25-197.39 imes 0.75 [/latex].

[latex] R_{D}=446.1 N, R_{E}=382.94 N [/latex].

Taking components of forces in horizontal and vertical directions, we have

[latex] 80+50 \cos heta_{1}+40 \cos heta_{2}=0 [/latex].

[latex] 50 \sin heta_{1}+40 \sin heta_{2}=0 [/latex].

Solving we get,

[latex] heta_{1}=-24.14^{\circ}, heta_{2}=149.25^{\circ} [/latex].

Question 12.18: A 4 m long shaft carries three pulleys, two at its ends and ...

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