Engineering in Kenya

# Center of Mass in Physics

Posted by on Jan 25, 2018 in Physics | 0 comments

The Center of Mass in Physics, also known as mass center or barycenter is the point of an object where all of its mass acts. In the case of a stationary body, the position of the Center of Mass in Physics or mass center is fixed in relation to the body itself. In the case of a distribution of masses in free space, such as a bullet from a shotgun or planets in the Solar System, the position of the Center of Mass in Physics or the mass center is a point in space that may not correspond to the position of the individual mass.

The use of the Center of Mass in Physics often allows the use of simplified equations of motion, as it is a convenient reference point for many principle calculations in physics, such as moments of inertia or the angular momentum. In many of its applications, such as orbital mechanics, an object can be replaced by a point mass located at the mass center for the sole purpose of analysis. Engineering in Kenya has more articles.

Mass in Physics

# Location of the Center of Mass in Physics

The term Center of Mass in Physics is often used in place of center of gravity, but they are different in pure physics concepts. They align/ coincide in a uniform gravitational field, but at a place where gravity is not uniform, center of gravity then refers to the mean location of the point at which the gravitational force acts on a body. This results in the small but non-negligible gravitational torque that must not be forfeited in the operation of artificial satellites.

The Center of Mass in Physics of an object does not always coincide with its geometric center, and this property can be well exploited. Engineers try to design a sports car’s Center of Mass in Physics to be as low as possible to make the car handling better and generally safer. When sky divers perform a Fosbury Flop, they bend their bodies in such a way that it is possible for the diver to clear the bar while the center of mass does not.

The center of momentum frame on the other hand is an inertial frame defined as the inertial frame in which the Center of Mass in Physics of a system is at rest. A certain center of momentum frame in which the Center of Mass in Physics is not only at rest, but also at the origin of the Cartesian/ coordinate system, is sometimes termed the center of mass frame, or the center of mass coordinate system.

The Center of Mass in Physics of a system of particles is defined as the mean of their positions, weighted by their masses. If an object has a uniform density, then its Center of Mass in Physics is the same as the geometric centre of its shape. E.g.

• The Center of Mass in Physics of a two particle system lies on the line connecting the particles, or more precisely, their centers of mass. The Center of Mass in Physics is closer to the more massive object.
• The center of mass of a uniform ring is at the center of the hollow of the ring. This is one of the examples of those objects with their Center of Mass in Physics outside the material that makes them up.
• The center of mass of a solid triangle of uniform density lies on all its three medians and therefore at centroid, which is the mean of its three vertices.
• The center of mass of a rectangle of uniform density is at the intersection of its two diagonals.
• In a symmetrical sphere-shaped body, the Center of Mass in Physics is at its geometric center. This applies to the Earth with approximation since its density varies considerably, but it mainly depends on the depth and less on the longitude and latitude coordinates.
• Generally, for any symmetrical body, its Center of Mass in Physics will be at a fixed point of that symmetry.

## History of the Center of Mass in Physics

The concept of Center of Mass in Physics was first introduced by the famous Greek mathematician, physicist, and engineer Archimedes. Archimedes showed that torque exerted on a lever by weights resting at different points on the lever is just the same as what it would be if all of the weights were moved from their different positions to a single point; the Center of Mass in Physics. In his work on floatation, he demonstrated that the orientation of the object is the one that shifts its center of mass to as low as possible. He formulated mathematical techniques for locating centers of mass of objects with uniform density and of various well-defined shapes, in particular a hemisphere, a triangle, and a frustum.

### Conclusion on Center of Mass in Physics; Derivation

The following equations in motion assume that there is a system of particles controlled by external and also internal forces. An internal force is defined as a force caused by the interaction of the particles within their system. An external force is defined as a force that originates from the outside of the system, and acts on one or more particles within the system. The external force in this case need not be due to a uniform field.

For any system which has no external forces, the Center of Mass in Physics moves with constant velocity. This applies to all systems with classical internal forces, including electric fields, magnetic fields, chemical reactions etc. Formally, this applies to any internal forces that satisfy the weaker form of Newton’s third law.

The total momentum for any system of particles is given by;

P = Mv;

where M indicates the mass and v the velocity of the Center of Mass in Physics. This velocity of the Center of Mass in Physics can be calculated by taking the time derivative of the actual position of the center of mass.

An analogue to Newton’s second law is;

F = Ma;

where F indicates the sum of all the external forces acting on the system, and a indicates the acceleration of the Center of Mass in Physics.

Letting the total internal force of the system;

F = MR

where M is the mass of the system and R is a vector yet to be defined, since:

P = ƩMiRi

and

F = P

then

R = 1/M (ƩMiRi)

Therefore, we have a vector-related definition for Center of Mass in Physics in terms of the total forces in the system.