Chapter 12 Equilibrium and Elasticity I Equilibrium Definition Requirements Static equilibrium II Center of gravity III Elasticity Tension and compression Shearing Hydraulic stress I Equilibrium Definition An object is in equilibrium if The linear momentum of its center of mass is constant Its angular momentum about its center of mass is constant Example block resting on a table hockey puck sliding across a frictionless surface with constant velocity the rotating blades of a ceiling fan the wheel of a bike traveling across a straight path at constant speed Static equilibrium P 0 L 0 Objects that are not moving either in TRANSLATION or ROTATION Example block resting on a table 1 Stable static equilibrium If a body returns to a state of static equilibrium after having been displaced from it by a force marble at the bottom of a spherical bowl Unstable static equilibrium A small force can displace the body and end the equilibrium 1 Torque about supporting edge by Fg is 0 because line of action of Fg passes through rotation axis domino in equilibrium 2 Slight force ends equilibrium line of action of Fg moves to one side of supporting edge torque due to Fg increases domino rotation 3 Not as unstable as 1 in order to topple it one needs to rotate it beyond balance position in 1 Requirements of equilibrium dP 0 P cte Fnet dt L cte net dL 0 dt Balance of forces translational equilibrium Balance of torques rotational equilibrium Vector sum of all external forces that act on body must be zero Vector sum of all external torques that act on the body measured about any possible point must be zero Balance of forces Fnet x Fnet y Fnet z 0 Balance of torques net x net y net z 0 2 II Center of gravity Gravitational force on extended body vector sum of the gravitational forces acting on the individual body s elements atoms cog Body s point where the gravitational force effectively acts This course initial assumption The center of gravity is at the center of mass If g is the same for all elements of a body then the body s Center Of Gravity COG is coincident with the body s Center Of Mass COM Assumption valid for every day objects g varies only slightly along Earth s surface and decreases in magnitude slightly with altitude Proof Each force Fgi produces a torque i on the element of mass about the origin O with moment arm xi r F i xi Fgi net i xi Fgi i i xcog Fg xcog Fgi net i xcog Fgi xi Fgi xcog mi g i xi mi g i xcog mi xi mi i xcog i i i i i 1 xi mi xcom M i 3 III Elasticity Branch of physics that describes how real bodies deform when forces are applied to them Real rigid bodies are elastic we can slightly change their dimensions by pulling pushing twisting or compressing them Stress Deforming force per unit area Shearing stress Strain Unit deformation Hydraulic stress Tensile stress associated with stretching Elastic modulus describes the elastic behavior deformations of objects as they respond to forces that act on them Stress Elasticity Modulus x Strain 1 Stress cte x Strain Recovers original dimensions when stress removed 2 Stress yield strength Sy specimen becomes permanently deformed 3 Stress ultimate strength Su specimen breaks Tension and compression Stress F A F force applied perpendicular to the area A of the object 4 Strain L L fractional change in length of the specimen Stress Young s modulus x Strain F L E A L Units of Young modulus F m2 Shearing Stress F A Strain x L F force in the plane of the area A F x G A L fractional change in length of the specimen Stress Shear modulus x Strain Hydraulic stress Stress Fluid pressure p F A p B V V Hydraulic Stress Bulk modulus x Hydraulic compression Strain V V 5
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