PAP111_Lecture14 - PAP111 Lecture14 TheCenterofGravity...

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PAP 111  Mechanics and Relativity  Lecture 14 Static Equilibrium and Elasticity The Conditions for Equilibrium The Center of Gravity Elastic Properties of Solids
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Equilibrium Up to this point, our study of equilibrium is limited to the  following: Such a situation is, however, insufficient when we deal with the  equilibrium of a rigid body.  The object is at rest or that its center of mass moves  with constant velocity relative to the observer.
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Effects of Force  F  on a Rigid  Object The force  F  which acts at  point  P  of the rigid object has  two effects: Translational acceleration of  the object. Rotational acceleration of  the object due to the torque  τ r  x  F , where  r  is the  position vector from  O  to  P . Note that in the case when  an object is modeled as a  particle, the force  F  only has  an effect on translation.
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The Conditions for Equilibrium In order for the rigid object to be in equilibrium, two necessary  conditions are required: Translational equilibrium: the resultant external force must be zero. -- The center of mass of the object has zero acceleration when viewed  from an inertial frame. Rotational equilibrium: the resultant external torque about  any  axis  must be zero. -- The angular acceleration of the rigid object about any axis is zero. = 0 F = 0 τ (16.1) (16.2)
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Equations for Equilibrium We will restrict our discussion to situations in which all the forces lie in  the  x - y  plane. Forces whose vector representation are in the same plane are said to be  coplanar. This consideration leads to three equations: Two come from balancing the forces in the  x  and  y  direction in Eq. (16.1) The third come from (16.2): the net torque about a perpendicular axis  through any point in the  x - y  plane must be zero. 0 0 = = y x F F = 0 z τ
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Question: Zero Net Torque about  any other Axis Show the following: If an object is in translational equilibrium (Eq. 16.1)  and the net torque is zero about one axis, then the  net torque must be zero about any other axis (Eq.  16.2).
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Static Equilibrium In general, Eq. (16.1) and (16.2) imply dynamic  equilibrium : The center of mass of the object moves with constant  velocity.
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