MHB What is the center of mass and how is it related to balancing an object?

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The center of mass is defined as the point at which an object can be balanced, with forces on either side being equal. It is often referred to as the center of gravity. Understanding the center of mass is crucial in calculus, particularly in Calculus 3, where it involves complex concepts like moment of inertia and radius of gyration. The textbook emphasizes that problems related to these topics require rigorous application of iterated integrals. Mastery of the center of mass is essential before progressing to triple integrals.
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In simple terms, what exactly is the center of mass as taught in Calculus 3?
 
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Imagine being able to balance an object from a needle supporting a single point inside the object. The "center of mass" is precisely that point. Any force on one side is balanced by a force on the other side. "Center of mass" is sometimes called the "center of gravity".
 
HallsofIvy said:
Imagine being able to balance an object from a needle supporting a single point inside the object. The "center of mass" is precisely that point. Any force on one side is balanced by a force on the other side. "Center of mass" is sometimes called the "center of gravity".

The chapter in my Calculus 3 textbook does not have too many questions concerning the Center of Mass, Moment of Inertia and Radius of Gyration. The authors of the textbook state that each question involves iterated integrals and thus can be very intense and rigorous. I will post one or two questions for each topic. I want to clearly grasp this material before moving on to triple integrals.
 
Thread 'Problem with calculating projections of curl using rotation of contour'

Thread 'Problem with calculating projections of curl using rotation of contour'

Hello! I tried to calculate projections of curl using rotation of coordinate system but I encountered with following problem. Given: ##rot_xA=\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}=0## ##rot_yA=\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x}=1## ##rot_zA=\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y}=0## I rotated ##yz##-plane of this coordinate system by an angle ##45## degrees about ##x##-axis and used rotation matrix to...
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