Lesson_Text_3.4 - 1 Lesson 3.4 Motion of the Center of Mass...

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1 Lesson 3.4 Motion of the Center of Mass 1. Center of Mass The equations of motion we developed so far, all describe the motion of point particles. More complicated objects are made up of large number of point particles. The motion of such a complicated object, like a foot ball, can be described by the motion of all the particles that make up this object. But all these particles are at slightly different positions and therefore, to describe the motion of the football, we have to describe the motion of every individual particles that make up the football. The concept of the center of mass is used to simplify this situation. Fig. 1 below shows two point particles of masses m 1 and m 2 situated at (x 1 , y 1 ) and (x 2 , y 2 ) respectively. The motion of these two particles is the same as the motion of a particle of mass (m 1 + m 2 ) placed at the point (x, y) somewhere between the two masses. mass m 1 placed at (x 1 , y 1 ) mass m 2 placed at (x 2 , y 2 ) (x, y) center of mass Fig. 1 The center of mass of m 1 and m 2 is at (x, y) This point (x, y) is called the center of mass of the two masses m 1 and m 2 . The center of mass of a system of particles is the point where the masses of all the particles of the system is assumed to be concentrated for the purpose of studying their linear motion.
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2 2. A formula for center of mass (CM) Consider a system that consists of two masses m 1 and m 2 . If a mass m 1 kg is situated at (x 1 , y 1 ) and a mass m 2 kg is situated at (x 2 , y 2 ), then the position of their center of mass (x cm , y cm ) is given by: 1 1 2 2 1 1 2 2 1 2 1 2 and cm cm m x m x m y m y x y m m m m + + = = + + This equation can be extended to systems that consist of many particles as: 1 1 2 2 3 3 1 2 3 and where ......... and ......... i i i i i i cm cm i i i i i i n n i i n i m x m y X Y m m m x m x m x m x m x m m m m m = = = + + + = + + + and The center of mass of a uniform regular object is at its geometric center. For example, the center of mass of a rectangular metal sheet is at its center. If you throw this metal sheet in the air, its motion will be the same as the motion of a point mass placed at its center of mass. . . Fig 2a center of mass is at the geometric center Fig. 2b Weight acts through the center of gravity If you now consider the weight of the particles of the system, there will be a point through which the resultant of the weight of all individual particles of
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3 the system may be assumed to be acting. This point is the same as the center of mass, but when we consider weight, we call it the center of gravity. The center of gravity of a system of particles is the point at
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This note was uploaded on 05/01/2011 for the course PHY 2048 taught by Professor George during the Fall '10 term at Edison State College.

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Lesson_Text_3.4 - 1 Lesson 3.4 Motion of the Center of Mass...

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