![]() The point at which the lines of symmetry of the lamina intersect can also be called the geometric center. Similarly, if a uniform lamina has more than one geometrical line of symmetry, then the center of mass will lieĪt the intersection of these lines of symmetry. If a geometrical axis of symmetry of a uniform lamina exists, then the center of mass lies along that line of Property: The Center of Mass of a Symmetrical Lamina Let us consider why this is theĬonsider the following uniform elliptical lamina with focus points at □ and □. This is because when we are considering a uniform lamina, the two are equal. When we are locating the center of mass of a uniform lamina, we need to identify the geometric center of the The □-coordinate value is necessarily 1 3 × 4 □. As the length of the side of □ □ □ in the □-direction is 4 5 the length of the side in the □-direction, This means that the position of the centroid is necessarily 1 3 of the length of its median,Īnd so the □-coordinate value is necessarily 1 3 × 5 □. The method of finding the centroid by intersection of medians could have been simplified greatlyĪs □ □ □ is a triangle, and we know that centroid divides the median in the ratio 2 ∶ 1. Therefore, the coordinates of the center of mass of □ □ □ are given by The value of □ must be multiplied by □, ![]() To obtain the □-coordinate of the point □ ( □, □ ), Substituting this value of □ into the equation of □ □ gives ![]() This can be rearranged as follows to determine □: There is a common factor of □ that can be eliminated to give These equations are equal, where □ and □ are the coordinates of the centroid of □ □ □: ![]() Values of □ and □ where the equation of line □ □ equals the equation of line □ □. The values of □ and □ can be determined algebraically by finding the The center of mass of the rod must be vertically above a part of the surface that provides ⃑ □, In order for □ and ⃑ □ to act to make the body be in equilibrium, ⃑ □, in the opposite direction to the weight of the rod, □, where □ will act along a line passing through the center of mass. If a rod is in contact with a surface, the surface will produce a reaction force, The same substitution of one force for a set of forces can be considered for the force of the weight of the rod.Ī rod resting on a surface exerts forces at every point of contact between the rod and the surface,īut the weight of a rod can be modeled as a single force that acts at the rod’s center of mass. Where □ is the length of the rod and □ is, therefore, at the midpoint of the rod.Ī force acting on the center of mass of a rod acts equivalently to a set of forces acting in the same direction at every point along the rod. In the case of a uniform rod, □ is given by This requires replacing the summation with the following integral: More realistic by allowing □ to tend to zero and, hence, The approximation of a system of □ particles can be made Where □ is the mass of the particle of index □ and □ is theĭistance from the origin of a coordinate system of the particle of index □. The position of □, the center of mass of a one-dimensional system of particles, Definition: The Center of Mass of a One-Dimensional System of Particles
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