Ch12 - Chapter 12: Static Equilibrium Conditions for...

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Chapter 12: Static Equilibrium Conditions for equilibrium ± First condition for equilibrium • When a particle is in equilibrium (no acceleration), in an inertial frame of reference, the vector sum of all the forces acting on the particle is zero. A frame of reference where Newton’s first law is valid = 0 F r = = = 0 , 0 , 0 z y x F F F The sum includes external forces only.
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Conditions for equilibrium (cont’d) ± Second condition for equilibrium • For an extended body to be in equilibrium the body must have no tendency to rotate. 0 / , 0 = = dt L d L r r about any point in the body = 0 τ r about any point in the body • Note that the above conditions apply to a rigid body in uniform translational motion or to constant rotation (if L is constant) as well. But in the following we will mostly consider situations in which a rigid body is at rest, i.e., in a static equilibrium. • The sum of the torques due to all external forces acting on the body with respect to any specified point must be zero.
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Center of gravity ± The center of gravity (= the center of mass) • For a collection of particles with masses and coordinates , the coordinates of the center of mass are given by: i m i i i z y x , , = ii i i i cm m x m x / = i i i cm m y m y / ± Definition of the center of mass • The gravitational torque about any point can be found by assuming that all the weight of the body acts at its center of gravity Although the gravity changes with the altitude, if we can ignore this change, this statement is valid. = i i i cm m z m z / = i i i cm m r m r / r r The center of gravity coincides with the center of mass when the gravitational field is uniform.
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Center of gravity (cont’d) ± Newton’s third law • For a collection of particles with masses and the center of mass coordinates , Newton’s third law is given by: i m cm cm cm z y x , , = = = = ii x i x i x i i i x net x M a m a a m f F , , , = = = = y i y i y i i i y net y M a m a a m f F , , , a M F net r r = = = = = z i z i z i i i z net z M a m a a m f F , , , M a a i r r = For a rigid body
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Center of gravity (cont’d) ± Total torque i m i r r z x y O cg=cm ?
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Ch12 - Chapter 12: Static Equilibrium Conditions for...

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