Luckily, most unknowns in equilibrium are linear terms, except for unknown angles. Note that the adjective “linear” specifies that the unknown values must be linear terms, which means that each unknown variable cannot be raised to a exponent, be an exponent, or buried inside of a \(\sin\) or \(\cos\) function. It is balanced when the beam remains level.\) \(y\) and \(z\) directions, you could be facing up to six equations and six unknown values.įrequently these simultaneous equation sets can be solved with substitution, but it is often be easier to solve large equation sets with linear algebra. The system is in static equilibrium when the beam does not rotate. Here, the free-body diagram for an extended rigid body helps us identify external torques.įigure 12.9 In a torque balance, a horizontal beam is supported at a fulcrum (indicated by S) and masses are attached to both sides of the fulcrum. The reason for this is that in analyzing rotation, we must identify torques acting on the body, and torque depends both on the acting force and on its lever arm. This does not hold true in rotational dynamics, where an extended rigid body cannot be represented by one point alone. In translational dynamics, a body is represented as its CM, where all forces on the body are attached and no torques appear. Also note that a free-body diagram for an extended rigid body that may undergo rotational motion is different from a free-body diagram for a body that experiences only translational motion (as you saw in the chapters on Newton’s laws of motion). Without the correct setup and a correct diagram, you will not be able to write down correct conditions for equilibrium. Note that setting up a free-body diagram for a rigid-body equilibrium problem is the most important component in the solution process. Also, you may independently check for your numerical answers by shifting the pivot to a different location and solving the problem again, which is what we did in Figure. If they do not, then use the previous steps to track back a mistake to its origin and correct it. Your final answers should have correct numerical values and correct physical units.
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