Sample Examination #1 Mechanical （样品检查# 1机械）.pdfVIP

Sample Examination #1 Closed book and notes. Calculators permitted For full marks you must a) insert relevant diagrams, b) explain in a few words what you are doing, c) present solutions neatly, d) indicate units where relevant, e) carry 3 significant figures. 1. The force F has a magnitude of 80 lb and acts within the volume shown. Determine the magnitudes of the x , y , and z components of F. (15 points) Write down the direction cosines first, and use them to find the magnitudes. We have α, β, but no γ yet, so start by finding γ. 2 2 2 cos α + cos β +cos γ = 1 2 ∴ (0.25 + 0.5 + cos γ) = 1 ∴ γ = 60° also Now magnitudes are Fx = 80 cos 60°= 40 lb F = 80 cos 45° = 56.6 lb y Fz = 80 cos 60°= 40 lb 2. Determine the magnitude of the projected component of force F along the pole. (15 points) This is one of the uses of the dot product ! We need 2 vectors and only have 1: so we need a unit vector along AO? AO = [-3i + 2j -6k] and its magnitude is 32 + 22 + 62 = 7 so a unit vector along AO is = 1/7[-3i + 2j -6k] and the dot product is F. AÔ = -20x-3/7 + 50x2/7 - 10x-6/7 = 60/7 + 100/7 + 60/7 =31.4 lb 3. Determine the stretch in each of the two springs required to hold the 20 kg crate in the equilibrium position shown. Each spring has an unstretched length of 2m and a stiffness of k = 300N/m. (20 points) 6i + 4j + 12k FOC = FOC = 3/7 FOCi + 2/7 FOCj + 6/7 FOCk 2 2 2