![]() Levers are the most basic machines that are used to do some work with minimal effort.of negligible mass) rod pivoted at a point along its length. An ideal lever is essentially a light (i.e.If they are at different locations, we use the sum of the moments equation and the distances of the people to determine the size of the reaction forces. ![]() The two reaction forces are equivalent because the forces on top are balanced evenly between the reaction forces. ![]() If we only consider the y (vertical) direction, the 200 lbs pushing down on the beam must be balanced by the reaction forces pushing up. Here is a visual example of using the equilibrium equations: Source: Engineering Mechanics, Jacob Moore, et al. Source: Engineering Mechanics, Jacob Moore, et al. The number of unknowns that you will be able to solve for will again be the number or equations that you have. Once you have your equilibrium equations, you can solve these formulas for unknowns. All moments will be about the z axis for two dimensional problems, though moments can be about x, y and z axes for three dimensional problems. To write out the moment equations simply sum the moments exerted by each force (adding in pure moments shown in the diagram) about the given point and the given axis (x, y, or z) and set that sum equal to zero. Remember that any force vector that travels through a given point will exert no moment about that point. Any point should work, but it is usually advantageous to choose a point that will decrease the number of unknowns in the equation. To do this you will need to choose a point to take the moments about. Next you will need to come up with the the moment equations. Your first equation will be the sum of the magnitudes of the components in the x direction being equal to zero, the second equation will be the sum of the magnitudes of the components in the y direction being equal to zero, and the third (if you have a 3D problem) will be the sum of the magnitudes in the z direction being equal to zero. Once you have chosen axes, you need to break down all of the force vectors into components along the x, y and z directions (see the vectors page in Appendix 1 page for more details on this process). If you choose coordinate axes that line up with some of your force vectors you will simplify later analysis. These axes do need to be perpendicular to one another, but they do not necessarily have to be horizontal or vertical. Next you will need to choose the x, y, z axes. In the free body diagram, provide values for any of the known magnitudes, directions, and points of application for the force vectors and provide variable names for any unknowns (either magnitudes, directions, or distances). This diagram should show all the force vectors acting on the body. $$\sum\vec M=0\\\sum M_x=0\:\sum M_y=0\:\sum M_z=0$$ Finding the Equilibrium Equations:Īs with particles, the first step in finding the equilibrium equations is to draw a free body diagram of the body being analyzed. We break the moments into three component equations We break the forces into three component equations The second set of three equilibrium equations states that the sum of the moment components about the x, y, and z axes must also be equal to zero. The body may also have moments about each of the three axes. There are three equilibrium equations for force, where the sum of the components in the x, y, and z direction must be equal to zero. If we look at a three dimensional problem we will increase the number of possible equilibrium equations to six. ![]() The one moment vector equation becomes a single moment scalar equation. The sum of each of these will be equal to zero.įor a two dimensional problem, we break our one vector force equation into two scalar component equations. This means that a rigid body in a two dimensional problem has three possible equilibrium equations that is, the sum of force components in the x and y directions, and the moments about the z axis. In a two dimensional problem, the body can only have clockwise or counter clockwise rotation (corresponding to rotations about the z axis). This means that our vector equation needs to be broken down into scalar components before we can solve the equilibrium equations. The addition of moments (as opposed to particles where we only looked at the forces) adds another set of possible equilibrium equations, allowing us to solve for more unknowns as compared to particle problems. For a rigid body in static equilibrium, that is a non-deformable body where forces are not concurrent, the sum of both the forces and the moments acting on the body must be equal to zero.
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