Lesson 1 - Center of Mass PDF

Lesson 1 Center of Mass Advanced General Physics Physics 1/2 Science, Third Quarter Technology, Engineering, and Mathematics The Roman architect Vitruvius (circa 90 BCE - 20 BCE) mathematically characterized the ideal proportions of the human body in his...

Lesson 1 Center of Mass Advanced General Physics Physics 1/2 Science, Third Quarter Technology, Engineering, and Mathematics The Roman architect Vitruvius (circa 90 BCE - 20 BCE) mathematically characterized the ideal proportions of the human body in his book De Architectura. 2 It inspired Leonardo da Vinci to draw “The Vitruvian Man.” It showed the exemplary biological symmetry of the human body. 3 Da Vinci drew an outer circle to show that a body’s center of mass stays affixed in space, even as the circle rotates. 4 In this lesson, you will learn how to identify the center of mass of both symmetrical and asymmetrical objects, and relate it to the stability of a physical system. 5  Learning Objectives At the end of the lesson, you should be able to do the following: Explain the concept of the center of mass. Distinguish an object’s center of mass from its geometric center. Solve problems related to a system’s center of mass. 7  The Centroid and the Center of Mass It is where the system’s total mass is concentrated when the system experiences external forces. It is a point or position that varies according to the composition, arrangement, or movement of an object or system. 8  The Centroid and the Center of Mass For rigid objects (circles, triangles, and rectangles), the center of mass is the midpoint of the system with respect to its uniform density. 9  The Centroid and the Center of Mass The dot in the middle is the geometric center of the plane or the centroid. Centroid is the arithmetic average of all Centroids in 2-D objects the points in a plane. 10  The Centroid and the Center of Mass For rigid homogeneous objects or those with uniform density, the center of mass and the geometric center Geometric centers of some (centroid) are the same. symmetrical objects 11  The Centroid and the Center of Mass The center of mass (measured in meters) is a distinct point in an object or system distinguished as the average of the individual masses factored by their distances from the point of reference. It is the average mass-weighted position of the particles in the system. 12  The Centroid and the Center of Mass Should the center of mass always be located at a physical point on or within the object? Where is the center of mass of a doughnut? 14  The Centroid and the Center of Mass For a toroid/torus (doughnut-shaped geometry), both the centroid and the center of mass are found right at the center of the empty space in the middle. The center of mass lies in the object’s axis of symmetry. 15  Calculating the Center of Mass The system’s center of mass is the point with coordinates (xcm, ycm) and is determined as follows: 16 Centroids from a reference line of Common 2D Shapes  Let’s Practice! A hanging light fixture contains four strings of lights, as follows: (1) a 0.33 kg light at 0.001 m, (2) a 0.45-kg light at 0.02 m, (3) a 0.77 kg light at 0.08 m, and (4) a 0.90-kg light at 1.1 m. Determine the hanging light fixture’s center of mass. 𝒎𝟏 𝒙𝟏 + 𝒎𝟐 𝒙𝟐 + 𝒎𝟑 𝒙𝟑 + 𝒎𝟒 𝒙𝟒 𝒎𝟏 𝒚𝟏 + 𝒎𝟐 𝒚𝟐 + 𝒎𝟑 𝒚𝟑 + 𝒎𝟒 𝒚𝟒 𝒙= 𝒚= 𝒎𝟏 + 𝒎𝟐 + 𝒎𝟑 + 𝒎𝟒 𝒎𝟏 + 𝒎𝟐 + 𝒎𝟑 + 𝒎𝟒 𝟎. 𝟑𝟑 𝟎. 𝟎𝟎𝟏 + 𝟎. 𝟒𝟓 𝟎. 𝟎𝟐 + 𝟎. 𝟕𝟕 𝟎. 𝟎𝟖 + 𝟎. 𝟗(𝟏. 𝟏) 𝒙= 𝟎. 𝟑𝟑 + 𝟎. 𝟒𝟓 + 𝟎. 𝟕𝟕 + 𝟎. 𝟗 𝒙 = 𝟎. 𝟒𝟑𝟑 𝒎 𝒂𝒍𝒐𝒏𝒈 𝒙 − 𝒂𝒙𝒊𝒔 18  Let’s Practice! A system containing two particles has a center of mass along the horizontal axis at x = 3.5 m. If the first particle is at 0 m from the origin and the second particle has a mass of 0.44 kg at x = 12.0 m, what is the mass of the first particle? 𝒎𝟏 𝒙𝟏 + 𝒎𝟐 𝒙𝟐 + 𝒎𝟑 𝒙𝟑 + ⋯ + 𝒎𝒏 𝒙𝒏 𝒙= 𝒎𝟏 + 𝒎𝟐 + 𝒎𝟑 + … + 𝒎𝒏 𝒎𝟏 𝟎 + 𝟎. 𝟒𝟒(𝟏𝟐) 𝟑. 𝟓 = 𝒎𝟏 + 𝟎. 𝟒𝟒 𝟑. 𝟓𝒎𝟏 + 𝟑. 𝟓 𝟎. 𝟒𝟒 = 𝟎 + 𝟎. 𝟒𝟒(𝟏𝟐) 𝒎𝟏 = 𝟏. 𝟎𝟔𝟗 𝒌𝒈 19  Let’s Practice! Three irregular blocks contain the following masses and C-M coordinates: (1) 0.56 kg (0.25 m,.45 m); (2) 0.67 kg (.15 m, -0.37 m); and (3).86 kg (-0.21 m, 0.77 m). Identify the coordinates of the center of mass of the three-block system. 𝒎𝟏 𝒙𝟏 + 𝒎𝟐 𝒙𝟐 + 𝒎𝟑 𝒙𝟑 + 𝒎𝟒 𝒙𝟒 𝒎𝟏 𝒚𝟏 + 𝒎𝟐 𝒚𝟐 + 𝒎𝟑 𝒚𝟑 + 𝒎𝟒 𝒚𝟒 𝒙= 𝒚= 𝒎𝟏 + 𝒎𝟐 + 𝒎𝟑 + 𝒎𝟒 𝒎𝟏 + 𝒎𝟐 + 𝒎𝟑 + 𝒎𝟒 𝟎. 𝟓𝟔 𝟎. 𝟐𝟓 + 𝟎. 𝟔𝟕 𝟎. 𝟏𝟓 + 𝟎. 𝟖𝟔 −𝟎. 𝟐𝟏 𝟎. 𝟓𝟔 𝟎. 𝟒𝟓 + 𝟎. 𝟔𝟕 −𝟎. 𝟑𝟕 + 𝟎. 𝟖𝟔 𝟎. 𝟕𝟕 𝒙= 𝒚= 𝟎. 𝟓𝟔 + 𝟎. 𝟔𝟕 + 𝟎. 𝟖𝟔 𝟎. 𝟓𝟔 + 𝟎. 𝟔𝟕 + 𝟎. 𝟖𝟔 𝒙 = 𝟎. 𝟎𝟐𝟗 𝒎 𝒚 = 𝟎. 𝟑𝟏𝟗 𝒎 𝑁𝑜𝑡𝑒: 𝑇ℎ𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑠𝑖𝑔𝑛 𝑜𝑛 𝑡ℎ𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑖𝑚𝑝𝑙𝑦 𝑡ℎ𝑎𝑡 𝑖𝑡 𝑖𝑠 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡 𝑠𝑖𝑑𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑒𝑟. 20