1. Mechanical Equilibrium By a definition, a system is in mechanical equilibrium if the sum of the forces acting on that system is equal to zero (i.c. E F = 0). If we combine this with Newton's 2" Law, we can make two conclusions: If EF = 0 = a = 0, then: i. The object remains stationary OR ii. The object is moving at a constant velocity Knowing an object is in equilibrium is a huge benefit to us. It allows us to solve complicated looking systems in a somewhat easy way. The steps to solving a problem involving equilibrium are: 1. Draw a Free Body diagram (this is non-negotiable) A good free body diagram makes these questions very easy . Label all your force vectors clearly! 2. Establish a coordinate system This is necessary for when we break our force vectors into components . On your free body diagram, any arrow that isn't exactly on an axis needs to be broken up 3. Set up your equations following from E F = 0 We will only be dealing with planar force in this class (2D cases) . Fill in the following equations _ F, = 0 and E F, = 0 This is why a good free body diagram is important! If it's clear and correct, all you need to do is look at the arrows going in the x direction (left and right) and the y direction (up and down)Problem 1 A 220 kg object hangs from a chain that is linked at point O to two other chains, one fastened to the ceiling and the other to the wall, as shown below. (10 pts) a) Draw a free body diagram (do it for the point () with 60% V appropriately labeled forces. b) Determine the tension in each of the three chains. T1 220 kgProblem 2 A delivery company has just unloaded a 57 kg crate full of home exercise equipment in your level driveway. Suppose you try to move the crate by pulling the rope at an angle of 40" above the horizontal. The coefficient of kinetic friction between the crate and the ground is 0.34. (15 pts) a) Draw a free body diagram with appropriately labeled forces. b) How much force do you need to pull the crate with so that it's moving at a constant velocity? 40* c) Determine the magnitude of the normal force. FFEProblem 3 A chair of mass 15 kg is sitting on the horizontal floor; the floor is not frictionless. You push on the chair with a force F = 30 N that is directed at an angle of 40 below the horizontal so that the chair slides at a constant velocity along the floor. (20 pts) a) Draw a free body diagram with appropriately labeled forces. b) What are the two components of the pushing force? c) What is the magnitude of the normal force? d) What is the magnitude of the friction force? e ) Determine the coefficient of kinetic friction.Problem 4 A 65 kg object is suspended by two cables, as shown below. (10 pts) 35' 48 a) Draw a free body diagram with appropriately labeled forces (draw it for the point where all the T1 Tz cables touch). b) Find the tension in each cord. T. m2. Static Equilibrium common misconception about equilibrium is that an object is not moving. BUT, as you saw from our definition of equilibrium on page 1, an object can be moving and still be in equilibrium. For example, a hild on a slide sliding down at a constant speed is in mechanical equilibrium. A bike tire being uspended but rotated is also in equilibrium. If we want to define an equilibrium where there is no motion and no rotation, then we need to add to our previous definition. rigid body is in static equilibrium if two conditions are satisfied: i . EF =0 ii. If = 0 As you can see. we need to impose that the sum of the torques on the object are also zero. This ensures here is no translational motion and rotational motion. Luckily for us, the addition of this constraint doesn't alter our problem-solving steps: 1. Draw a Free Body diagram This is crucial! No matter how complicated the problem seems, it can always be made simpler with a good, simple free body diagram 2. Establish a coordinate system 3. Establish an axis of rotation . This is important for the torque part 4. Set up your equations following from E F = 0 . EF = 0 and EFy = 0 5. Set up your equations following from Ef = 0 Elew = Etcew What determines if a torque is CW or CCW depends on where the axis of rotation is!Problem 1 The figure to the right shows a construction worker standing on a uniform beam. The beam has a mass of 85 kg and a length of L = 20 m. The construction work in the figure has a mass of 1 10 kg and is standing 5 m from the hinge (black dot). A cable (assume it's massless) is connected from the wall to the end of the beam and makes and angle of 0 = 30" (15 pts) a) Draw a free body diagram for the beam. Include all forces acting on the beam (you should have $ force vectors) b) Put the axis of rotation at the hinge. Which forces produce a torque about the hinge? Which forces do not? c) Write out the equation for equilibrium (equations (i) and (ii) on page 6) L d) Solve for the tension in the cableProblem 2 A 66 kg diver stands at the end of a 3 m diving board as shown below. The diving board has a uniform mass of 25 kg. The board is attached to a hinge (black dot) at the left end and rests on top of the right support. (15 pts) 1.5 m a) Draw a free body diagram for the beam. Include all forces acting on the beam (you should have 4 force vectors) b) Put the axis of rotation at the hinge. Which forces 3.0 m produce a torque about the hinge? Which forces do not? c) Write out the equation for equilibrium (equations (i) and (ii) on page 6) d) Solve for the vertical force exerted by the hinge and by the right supportProblem 3 A bowler holds a bowling ball (M = 7.3 kg) in the palm of their hand as shown. The mass of forearm is 2 Biceps kg. (20 pts) a) Draw a free body diagram for the forearm. Elbow Include all forces acting on the forearm (you contact should have 4 force vectors) point -4.0 cm Lower arm 15 cmi [forearm plus b) Put the axis of rotation at the elbow contact 35 cm hand) center point. Which forces produce a torque about the elbow? Which forces do not? c) Write out the equation for equilibrium (equations (i) and (ii) on page 6) d) What is the magnitude of the force of the biceps muscle on the forearm? What is the magnitude of the force of the biceps muscle if the forearm is lowered 30"? f) If the maximum force the biceps can handle is 1000 N. what's the heaviest weight you can hold if the forearm is in the horizontal position