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PROBLEM 1 (20 points) – Prob. 1 questions are all or nothing.

1A. Newton’s laws are the basis for mechanics, including the mechanics of systems in equilibrium. In order, please state Newton’s Three Laws of Motion using complete sentences and/or equations.

Newton’s 1st Law (2pts):

Newton’s 2nd Law (2pts):

Newton’s 3rd Law (2pts):

1B. Given T1  = 30kN-m and T2  = 10kN-m, determine the maximum shear stress due to torsion in

shaft BC. Determine the maximum shear stress due to torsion in shaft CD. Assume do   = 0.1m, di   = 0.025m,  and L = 10m. (6 pts)

1C. For  the  triangular  cross  section  shown  below,  set  up  but  do  not  integrate  the  equation  for calculating the y coordinate of the centroid. Be sure to clearly indicate all limits of integration as well as the function being integrated. (4 points)

1D. A punching machine is capable of delivering up to 440 kips of force. The machine was designed to punch 4in diameter circles out of 2014-T6 aluminum (E = 10.6x103  ksi, shear strength = 70 ksi).     What is the maximum thickness of aluminum plate which the machine can punch through? (4 points)

PROBLEM 2 (20 points) – Prob. 2 questions are all or nothing.

2A. A submerged flood gate is 5 feet in length and 3 feet in width (in/out of the page). The bottom of  the gate (point B) is 10 feet below the surface of the water. Answer the following questions about the flood gate. (4 pts)

Calculate the pressueat points A and B. Note that the specific gravity of water is 62.4 lbs/ft3 .

2B. A block, M1 , is resting on a horizontal surface with a coefficient of friction μ=0.288. A second block, M2 , is on a smooth ramp with an inclination of θ=60。. M2  has a mass of 10kg. They are connected with an inextensible cable which wraps partially around a drum. The drum has a coefficient of friction μ=0.387. (10 points)

Calculate the tensions in the upper cable segment (T1) and the lower cable segment (T2). Then calculate the minimum mass, M1 , to prevent motion.

2C. A cantilevered beam is shown below. The value of F is 100N. Replace the loading shown with an equivalent force (please write in vector form) and moment (please write in vector form) at point A.

Please note that this is an equivalent system problem. (6 pts)

PROBLEM 3 (20 points)

A massless beam OAB shown in the figure is rigidly connected to the wall (indicated as gray area). It also connects to cable AC with 300N in tension at point A, the other end of the cable C is located on the wall. In addition, a vertical downward force of 100N is applied at point B of the beam. The distance between O and A is 2m, and 1m between A and B.

3A. Using the figure below, complete the free body diagram of the beam (4pts)

3B. The tension in cable AC is given as 300N. Rewrite this tension in vector form as a magnitude times a unit vector. (5pts)

3C. Calculate the reaction forces at connecting point O. Express your answers in standard vector format. (5pts)

standard vector format. (5pts)

PROBLEM 4 (20 points)

The truss below is in static equilibrium and supports loads at joints C, and F. The support at K is a roller.

4A. Identify all zero-force members (no explanation is required). (4 pts)

4B. Draw the free body diagram of the truss on the figure provided below. (4pts).

4C. Compute the reaction forces on supports A and K. (4pts)

4D. Determine the forces in members CD and EF. Write your answers in the box below. Indicate if

the member is in tension or compression by circling the appropriate option. Show your work including the necessary free body diagrams. (8pts)

PROBLEM 5 (20 points)

Given: Beam ABCDE is loaded as shown and is held in static equilibrium by a pin support at A and a roller support at D.  The beam cross-section is an “inverted T-shape” and has the dimensions shown  with a second moment of inertia of Ix   = 33.3 in4    .  (Note – NA refers to the neutral axis.)

Find:

5a) Sketch a free-body diagram of the beam and determine the magnitudes of the reactions at A and

D. (5 pts)

5b) On the axes provided, sketch the shear-force and bending-moment diagrams of the beam. Please specify the magnitudes of the shear and moment at each of the transitionpoints. (8 pts

5c) In which segment(s) of the beam does pure bending occur? (1 pts)

5d) In the segment of the beam where pure bending exists, determine the magnitudes of the maximum tensile bending stress and the maximum compressive bending stress. (6 pts)