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om3 - hernandez(ejh742 statics and elasticity problems...

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hernandez (ejh742) – statics and elasticity problems – Turner – (58120) 1 This print-out should have 16 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points A cylindrical stone column of diameter 2 R = 1 . 35 m and height H = 2 . 69 m is transported in standing position by a dolly. a side view When the dolly accelerates or decelerates slowly enough, the column stands upright, but when the dolly’s acceleration magnitude exceed a critical value a c , the column top- ples over. (For a > + a c the column topples backward; for a < - a c the column toppes forward.) Calculate the magnitude of the critical ac- celeration a c of the dolly. The acceleration of gravity is 9 . 8 m / s 2 . Correct answer: 4 . 91822 m / s 2 . Explanation: Let : g = 9 . 8 m / s 2 , R = 0 . 675 m , and H = 2 . 69 m . In the non-inertial frame of the accelerating dolly, the column is subject to the horizontal inertial force vector F in = - mvectorg . Together, the gravity and the inertial force combine into the apparent weight force vector W app = m ( vectorg - vectora ) in the direction θ = arctan parenleftbigg a g parenrightbigg from the vertical. From the torque point of view, this appar- ent weight force applies at the center of mass of the column. The column is stable in the vertical position when the line of this force goes through the column’s base CM W app but when this line misses the base, the column topples over CM W app For the critical acceleration a c , the line goes through the edge of the base, hence the direc- tion of the apparent weight force must deviate from the vertical by the angle θ c where tan θ c = R h cm = 2 R H . Consequently, the critical acceleration of the dolly is a c = g tan θ c = g 2 R H = (9 . 8 m / s 2 ) 2 (0 . 675 m) 2 . 69 m = 4 . 91822 m / s 2 . 002 10.0 points The system shown in the figure is in equilib- rium. A 13 kg mass is on the table. A string attached to the knot and the ceiling makes an angle of 60 with the horizontal. The coeffi- cient of the static friction between the 13 kg mass and the surface on which it rests is 0 . 31.
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hernandez (ejh742) – statics and elasticity problems – Turner – (58120) 2 13 kg m 60 What is the largest mass m can have and still preserve the equilibrium? The accelera- tion of gravity is 9 . 8 m / s 2 . Correct answer: 6 . 98017 kg. Explanation: Let : M = 13 kg , m = 6 . 98017 kg , and θ = 60 . For the system to remain in equilibrium, the net forces on both M and m should be zero, so the tension in the rope has an upper bound value T max , where T max cos θ = μ M g (1) T max = μ M g cos θ = (0 . 31) (13 kg) (9 . 8 m / s 2 ) cos 60 = 78 . 9881 N .
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