Physics 1408-002 Principles of Physics Sung-Won Lee [email protected] Lecture 19 – Chapter 12 – March 26, 2009 Announcement I Lecture note is on the web Handout (6 slides/page) http://highenergy.phys.ttu.edu/~slee/1408/ *** Class attendance is strongly encouraged and will be taken randomly. Also it will be used for extra credits. HW Assignment #8 will be placed on MateringPHYSICS today, and is due by 11:59pm on Wendseday, 4/1 Chapter 12 Static Equilibrium; Elasticity and Fracture •! The Conditions for Equilibrium •! Solving Statics Problems •! Stability and Balance •! Elasticity; Stress and Strain •! Fracture •! Trusses and Bridges / Arches and DomesApproach to Statics •! In general, we can use the two equations!to solve any statics problem.!When choosing axes about which to calculate torque, the choice can make the problem easy....! !!= 0" !F = 0!Torque due to Gravity m4 m1 m2 m3 x4 Consider total external torque on a system of masses in a uniform gravitational field. 12-2 Solving Statics Problems A board of mass M = 2.0 kg serves as a seesaw for two children. Child A has a mass of 30 kg and sits 2.5 m from the pivot point, P (his center of gravity is 2.5 m from the pivot). At what distance x from the pivot must child B, of mass 25 kg, place herself to balance the seesaw? Assume the board is uniform and centered over the pivot.12-2 Solving Statics Problems At what distance x from the pivot must child B, of mass 25 kg, place herself to balance the seesaw? Assume the board is uniform and centered over the pivot. Hanging Lamp •! A lamp of mass M hangs from the end of plank of mass m and length L. One end of the plank is held to a wall by a hinge, and the other end is supported by a massless string that makes an angle with the plank. (The hinge supplies a force to hold the end of the plank in place.) –!What is the tension in the string? –!What are the forces supplied by the hinge on the plank? hinge M m L ! "Hanging Lamp... •! First use the fact that in both x and y directions: M m L/2 ! "Fx Fy L/2 Mg mg y x !F = 0!x: T cos ! + Fx = 0 y: T sin ! + Fy - Mg - mg = 0 !! Now use in the z direction. "!If we choose the rotation axis to be through the hinge then the hinge forces Fx and Fy will not enter into the torque equation: !!= 0"LMg +L2mg - LT sin!= 0Hanging Lamp... •! So we have three equations and three unknowns:"!T cos ! + Fx = 0 T sin ! + Fy - Mg - mg = 0 M m L/2 ! "Fx Fy L/2 Mg mg y x LMg +L2mg - LT sin!= 0which we can solve to find: T =M +m2!"#$%&gsin'Fx=! M +m2"#$%&'gtan(ˆiFy=12mgˆjIf the forces on an object are such that they tend to return it to its equilibrium position, it is said to be in stable equilibrium. 12-3 Stability and Balance If, however, the forces tend to move it away from its equilibrium point, it is said to be in unstable equilibrium. 12-3 Stability and Balance An object in stable equilibrium may become unstable if it is tipped so that its center of gravity is outside the pivot point. Of course, it will be stable again once it lands! 12-3 Stability and Balance People carrying heavy loads automatically adjust their posture so their center of mass is over their feet. This can lead to injury if the contortion is too great. 12-3 Stability and Balance Humans adjust their posture to achieve stability when carrying loads.Hooke’s law: the change in length (#l) is proportional to the applied force (F). 12-4 Elasticity; Stress and Strain 12-4 Elasticity; Stress and Strain •! All objects are deformable, i.e. it is possible to change the shape or size (or both) of an object through the application of external forces –! Sometimes when the forces are removed, the object tends to its original shape, called elastic behavior –! Large enough forces will break the bonds between molecules and also the object The change in length of a stretched object depends not only on the applied force, but also on its length, cross-sectional area and the material from which it is made. The material factor, E, is called the elastic modulus or Young’s modulus, and it has been measured for many materials. 12-4 Elasticity; Stress and Strain Elastic modulus=stressstrain12-4 Elasticity; Stress and Strain Stress is defined as the force per unit area. (F/A) Strain is defined as the ratio of the change in length to the original length. (#l/l0) Therefore, the elastic modulus (E) is equal to the stress divided by the strain: Young’s Modulus: Elasticity in Length •! Tensile stress is the ratio of the external force (F) to cross-sectional area (A) –! For both tension and compression •! The elastic modulus is called Young’s modulus •! SI units of stress: Pascals, [Pa] –! 1 Pa = 1 N/m2 •! The tensile strain is the ratio of the change in length to the original length Y =tensile stresstensile strain=FA!LLo=F LoA