Slide 1AnnouncementsSimilarity Between Linear and Rotational MotionsConditions for EquilibriumMore on Conditions for EquilibriumHow do we solve static equilibrium problems?Example for Mechanical EquilibriumExample for Mech. Equilibrium Cont’dEx. 9.8 for Mechanical EquilibriumElastic Properties of SolidsExample 9 – 7Example 9 – 7 cont’dEx. A Diving BoardFinger Holds Water in StrawYoung’s ModulusBulk ModulusExample for Solid’s Elastic PropertyDensity and Specific GravityFluid and PressureExample for PressureBuoyant Forces and Archimedes’ PrincipleMore Archimedes’ PrincipleMore Archimedes’ PrincipleExample for Archimedes’ PrincipleExample for Buoyant ForceFlow Rate and the Equation of ContinuityExample for Equation of ContinuityBernoulli’s PrincipleBernoulli’s Equation cont’dBernoulli’s Equation cont’dExample for Bernoulli’s EquationMonday, Dec. 6, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu1PHYS 1441 – Section 002Lecture #23Monday, Dec. 6, 2010Dr. Jaehoon Yu•Similarities Between Linear and Rotational Quantities•Conditions for Equilibrium•How to Solve Equilibrium Problems?•A Few Examples of Mechanical Equilibrium•Elastic Properties of Solids•Density and Specific GravityAnnouncements•The Final Exam–Date and time: 11am – 1:30pm, Monday Dec. 13–Place: SH103–Comprehensive exam•Covers from CH1.1 – what we finish Wednesday, Dec. 8•Plus appendices A.1 – A.8•Combination of multiple choice and free response problems •Bring your Planetarium extra credit sheet to the class next Wednesday, Dec. 8, with your name clearly marked on the sheet!Monday, Dec. 6, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu2Monday, Dec. 6, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu3Similarity Between Linear and Rotational MotionsAll physical quantities in linear and rotational motions show striking similarity.QuantitiesLinear RotationalMass Mass Moment of InertiaLength of motion Distance Angle (Radian)SpeedAccelerationForce Force TorqueWork Work WorkPowerMomentumKinetic Energy Kinetic Rotational IrvtD=DtqwD=DvatD=DtwaD=D Fur=mar τr=IαurW F d= �rr P =Fur⋅vrP221mvK 221IKRLMW t q= pur=mvr Lur=IωurMonday, Dec. 6, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu4Conditions for EquilibriumWhat do you think the term “An object is at its equilibrium” means? Fur∑=The object is either at rest (Static Equilibrium) or its center of mass is moving at a constant velocity (Dynamic Equilibrium). Is this it? When do you think an object is at its equilibrium?Translational Equilibrium: Equilibrium in linear motion The above condition is sufficient for a point-like object to be at its translational equilibrium. However for an object with size this is not sufficient. One more condition is needed. What is it? Let’s consider two forces equal in magnitude but in opposite direction acting on a rigid object as shown in the figure. What do you think will happen?CMddF-FThe object will rotate about the CM. Since the net torque acting on the object about a rotational axis is not0. For an object to be at its static equilibrium, the object should not have linear or angular speed. t =�r0CMv =0w =00Monday, Dec. 6, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu5More on Conditions for EquilibriumTo simplify the problem, we will only deal with forces acting on x-y plane, giving torque only along z-axis. What do you think the conditions for equilibrium be in this case? The six possible equations from the two vector equations turns to three equations.What happens if there are many forces exerting on an object?0F =�ur0t =�r0xF =�0zt =�Or5O’r’If an object is at its translational static equilibrium, and if the net torque acting on the object is 0 about one axis, the net torque must be 0 about any arbitrary axis.0yF =�Why is this true?Because the object is not moving, no matter what the rotational axis is, there should not be any motion. It is simply a matter of mathematical manipulation.ANDMonday, Dec. 6, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu6How do we solve static equilibrium problems?1. Select the object to which the equations for equilibrium are to be applied.2. Identify all the forces and draw a free-body diagram with them indicated on it with their directions and locations properly indicated3. Choose a convenient set of x and y axes and write down force equation for each x and y component with correct signs.4. Apply the equations that specify the balance of forces at equilibrium. Set the net force in the x and y directions equal to 0.5. Select the most optimal rotational axis for torque calculations Selecting the axis such that the torque of one or more of the unknown forces become 0 makes the problem much easier to solve.6. Write down the torque equation with proper signs.7. Solve the force and torque equations for the desired unknown quantities.Monday, Dec. 6, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu7Example for Mechanical EquilibriumA uniform 40.0 N board supports the father and the daughter each weighing 800 N and 350 N, respectively, and is not moving. If the support (or fulcrum) is under the center of gravity of the board, and the father is 1.00 m from CoG, what is the magnitude of the normal force n exerted on the board by the support?Since there is no linear motion, this system is in its translational equilibriumFDn1m xTherefore the magnitude of the normal force nDetermine where the child should sit to balance the system.The net torque about the fulcrum by the three forces are Therefore to balance the system the daughter must sitxFx∑0yFBM g-0FM g-DM g-n=mgMgMDF00.1mm 29.200.135080000.1 gMFxgMD0N11903508000.40 MBgMDgMFg0 gMB0n+ �Monday, Dec. 6, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu8Example for Mech. Equilibrium Cont’d Determine the position of the child to balance the system for different position of axis of rotation.Since the normal force is The net torque about the axis of rotation by all the forces are ThereforexnThe net torque can be rewritten What do we learn?No matter where the rotation axis is, net effect of the torque is identical.FDnMBgMFg MFg1m xx/2Rotational axis2/xgMB0gMgMgMDFB 2/00.1 xgMF2/xn 2/xgMD2/xgMB 2/00.1 xgMF 2/xgMgMgMDFB2/xgMDxgMgMDF 00.10mgMgMDF00.1mm 29.200.1350800Monday,
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