# Center of mass position and velocity relationship

The position of the center of mass of a system of two particles with mass m1 and This equation shows that the center of mass lies between the two masses, . net external force acting on the system is zero, the velocity of the center of mass. The center of mass is a position defined relative to an object or system of objects. It is the average position of all the parts of the system, weighted according to. position vector of the center of mass can be defined as: the same direction of the velocity. This is known as the momentum-kinetic energy relationship.

The plumb line method Figure 5 is also useful for objects which can be suspended freely about a point of rotation. An irregularly shaped piece of cardboard suspended on a pin-board is a good example of this.

The cardboard pivots freely around the pin under gravity and reaches a stable point. A plumb line is hung from the pin and used to mark a line on the object. The pin is moved to another location and the procedure repeated. The center of mass then lies beneath the intersection point of the two lines.

Plumb line method being used to find the center of mass of an irregular object. Center of mass and toppling stability One useful application of the center of mass is determining the maximum angle that an object can be tilted before it will topple over.

Figure 6a shows a cross section of a truck.

### Center of mass - Wikipedia

The truck has been poorly loaded with many heavy items loaded on the left-hand side. The center of mass is shown as a red dot. A red line extends down from the center of mass, representing the force of gravity. Gravity acts on all the weight of the truck through this line. Center of mass The motion of a rotating ax thrown between two jugglers looks rather complicated, and very different from the standard projectile motion discussed in Chapter 4.

Experiments have shown that one point of the ax follows a trajectory described by the standard equations of motion of a projectile. This special point is called the center of mass of the ax.

The position of the center of mass of a system of two particles with mass m1 and m2, located at position x1 and x2, respectively, is defined as Since we are free to define our coordinate system in whatever way is convenient, we can define the origin of our coordinate system to coincide with the left most object see Figure 9.

- Center of mass

The position of the center of mass is now Figure 9. Position of the center of mass in 1 dimension. This equation shows that the center of mass lies between the two masses, closest to the heavier mass. In general, for a system with more than two particles, the position of the center of mass will satisfy the following relation The definition of the center of mass in one dimension can be easily generalized to three dimensions or in vector notation For a rigid body, the summation will be replaced by an integral Suppose we are dealing with a number of objects.

The position of the center of mass of m1 and m2 is given by The position of the center of mass of m3 and m4 is given by The position of the center of mass of the whole system is given by This can be rewritten as Using the center of mass of m1 and m2 and of m3 and m4 we can express the center of mass of the whole system as follows Figure 9. Location of 4 masses. This shows that the center of mass of a system can be calculated from the position of the center of mass of all objects that make up the system.

For example, the position of the center of mass of a system consisting out of several spheres can be calculated by assuming that the mass of each sphere is concentrated in the center of that sphere its center of mass.

## What is center of mass?

The center of mass of an object does not need to lie within the body of that object for example: Sample Problem Figure 9. Let us call it object X. Locate the center of mass of object X. Symmetry arguments immediately tell us that the center of mass of object X is located on the x-axis.

Suppose the hole in object X is filled with a disk of radius R. The new object object C, Figure 9. The center of mass of this system consisting out of object X and object D can be easily calculated: This equation can be rewritten as For a homogeneous disk with density [rho] the masses of object X and D can be calculated Figure 9. The position of the center of mass of object X is given by Example Problem Figure 9.

The density of the rod is position dependent: Determine the location of the center-of-mass of the rod. Differentiating this equation with respect to time shows where vcm is the velocity of the center of mass and vi is the velocity of mass mi. The acceleration of the center of mass can be obtained by once again differentiating this expression with respect to time where acm is the acceleration of the center of mass and ai is the acceleration of mass mi.

### What is center of mass? (article) | Khan Academy

Using Newton's second law we can identify mi ai with the force acting on mass mi. This shows that This equation shows that the motion of the center of mass is only determined by the external forces. Forces exerted by one part of the system on other parts of the system are called internal forces. According to Newton's third law, the sum of all internal forces cancel out for each interaction there are two forces acting on two parts: Internal and External Forces acting on a System of Particles.

The previous equations show that the center of mass of a system of particles acts like a particle of mass M, and reacts like a particle when the system is exposed to external forces.

They also show that when the net external force acting on the system is zero, the velocity of the center of mass will be constant. Example Problem The center of mass of an exploding rocket will follow the trajectory of a projectile. The forces of the explosion are internal to the system, and the only external force acting on the system is the gravitational force.

Example Problem A ball of mass m and radius R is placed inside a spherical shell of the same mass m and inner radius 2R see Figure 9. The ball is released and moves back and forth before coming to rest at the bottom of the shell see Figure 9.

What is the displacement of the system? The only external forces acting on the system are the gravitational force and the normal force. Both act in the y-direction.

The x-component of the total external force acting on the system is zero.