1. A hunter decides to build a cheap gate for his hunting ground by cutting down

Question

1. A hunter decides to build a cheap gate for his hunting ground by cutting down a small oak tree, trimming off the branches, and attaching it to another tree as shown. The horizontal oak log is 12 ft long and 6 inches in diameter. The cable is 9 ft long. The point where the cable is attached to the tree (B) is 4 ft above the point where the log joins the tree (A). What is the optimal B-A height for which the tension in the cable is a minimum?

2. Determine the rate of precession for a gyroscope that consists of a rotating wheel spinning at 500 RPM. The wheel is a solid disk of diameter 3 inches that weighs ½ lb. The wheel precesses on an arm 2.5 inches in length.

Explanation / Answer

For the system to be in the equilibrium, net torque due to the weight of the oak log

Tension int he cable about the point A should be zero

Mass of the oak log m =V*p =(pir2l)p =3.14(6in(0.0254/1in))2(12ft)(0.3048m/1ft)(740kg/m3) =197.5kg

Weight of the oak W =mg =(197.5kg)(9.81m/s2) =1935.5N

Now assuming the tension in the cable whihc makes an angl theta with the horizontal

consider counter clockwise as positive

+(Tsintheta)(AC) -W(l/2) =0

(Tsinthea)(AC) =W(l/2)

T(4ft/9ft)(Sqrt(9ft)2-(4ft)2 =(1935.5N)(6ft)

T =3240.9N =3241N

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