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Calculate and Verify Center of Mass

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Terms you will need

  • center of massThe center of mass is a point in an object or system of objects where the entire mass can be considered to be concentrated for the purpose of analyzing motion or balance.
  • centroidThe centroid is the geometric center of a shape or object, where it can be thought of as the average position of all the points in the shape.
  • moment of inertiaThe moment of inertia is a measure of an object's resistance to changes in its rotational motion, depending on the mass distribution and the axis of rotation.
  • coordinate systemA coordinate system is a method used to describe the position of points in space using numbers, typically along axes such as x, y, and z.
  • x-coordinateThe x-coordinate is the first number in a set of coordinates that describes the position of a point along the horizontal axis in a coordinate system.
  • y-coordinateThe y-coordinate is the second number in a set of coordinates that describes the position of a point along the vertical axis in a coordinate system.

How this topic runs

  1. 1Start here
  2. 2How it works
  3. 3At competition depth

The center of mass is a fundamental concept in physics that describes the average position of the mass in a system. It is a point where the entire mass of the system can be considered to be concentrated for the purposes of analyzing motion and forces. Understanding the center of mass is essential for solving problems in mechanics, especially when dealing with systems composed of multiple objects or particles. This concept is crucial in various applications, including engineering, astronomy, and everyday physics, as it helps in predicting the behavior of objects under the influence of external forces and in determining the stability and balance of structures.

Figures (7)

Figure 9.6 Torque is the turning or twisting effectiveness of a force, illustrated here for door rotation on its hinges (as viewed from overhead). Torque has both magnitude and direction. (a) Counterclockwise torque is produced by this force, which means that the door will rotate in a counterclockwise due to FF. Note that r⊥ r⊥ is the perpendicular distance of the pivot from the line of action of the force. (b) A smaller counterclockwise torque is produced by a smaller force F′F′ acting at the same distance from the hinges (the pivot point). (c) The same force as in (a) produces a smaller counterclockwise torque when applied at a smaller distance from the hinges. (d) The same force as in (a), but acting in the opposite direction, produces a clockwise torque. (e) A smaller counterclockwise torque is produced by the same magnitude force acting at the same point but in a different direction. Here, θθ is less than 90º90º. (f) Torque is zero here since the force just pulls on the hinges, producing no rotation. In this case, θ=0ºθ=0º.
Figure 9.7 A force applied to an object can produce a torque, which depends on the location of the pivot point. (a) The three factors rr, FF, and θθ for pivot point A on a body are shown here—rr is the distance from the chosen pivot point to the point where the force FF is applied, and θθ is the angle between FF and the vector directed from the point of application to the pivot point. If the object can rotate around point A, it will rotate counterclockwise. This means that torque is counterclockwise relative to pivot A. (b) In this case, point B is the pivot point. The torque from the applied force will cause a clockwise rotation around point B, and so it is a clockwise torque relative to B.
Figure 9.8 Two children balancing a seesaw satisfy both conditions for equilibrium. The lighter child sits farther from the pivot to create a torque equal in magnitude to that of the heavier child.
Figure 9.2 This motionless person is in static equilibrium. The forces acting on him add up to zero. Both forces are vertical in this case.
Figure 9.3 This car is in dynamic equilibrium because it is moving at constant velocity. There are horizontal and vertical forces, but the net external force in any direction is zero. The applied force FappFapp between the tires and the road is balanced by air friction, and the weight of the car is supported by the normal forces, here shown to be equal for all four tires.
Figure 9.4 An ice hockey stick lying flat on ice with two equal and opposite horizontal forces applied to it. Friction is negligible, and the gravitational force is balanced by the support of the ice (a normal force). Thus, netF=0netF=0. Equilibrium is achieved, which is static equilibrium in this case.
Figure 9.5 The same forces are applied at other points and the stick rotates—in fact, it experiences an accelerated rotation. Here netF=0netF=0 but the system is not at equilibrium. Hence, the netF=0netF=0 is a necessary—but not sufficient—condition for achieving equilibrium.

It is where the system's entire mass can be thought of as being concentrated for analyzing motion and forces. This concept is important for understanding how objects move and balance, especially when dealing with multiple objects or parts. To picture it, imagine balancing an object on a single point — that point is the center of mass.

It helps predict how objects will behave under forces and how stable they are. Calculating it involves considering the mass and position of each part of the system.

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