Planar lamina

In mathematics, a planar lamina is a closed set in a plane of mass and surface density such that:

The center of mass of the lamina is at the point

where moment of the entire lamina about the x-axis and moment of the entire lamina about the y-axis.

Example 1.

Find the center of mass of a lamina with edges given by the lines , and , where the density is given as .

center of mass is at the point

Planar laminas can be used to determine moments of inertia, or center of mass.

Center of mass

In physics, the center of mass of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero or the point where if a force is applied causes it to move in direction of force without rotation. The distribution of mass is balanced around the center of mass and the average of the weighted position coordinates of the distributed mass defines its coordinates. All forces are equal and opposite. Calculations in mechanics are often simplified when formulated with respect to the center of mass.

In the case of a single rigid body, the center of mass is fixed in relation to the body, and if the body has uniform density, it will be located at the centroid. The center of mass may be located outside the physical body, as is sometimes the case for hollow or open-shaped objects, such as a horseshoe. In the case of a distribution of separate bodies, such as the planets of the Solar System, the center of mass may not correspond to the position of any individual member of the system.

Mass

In physics, mass is a property of a physical body. It is generally a measure of an object's resistance to changing its state of motion when a force is applied. It is determined by the strength of its mutual gravitational attraction to other bodies, its resistance to being accelerated by a force, and in the theory of relativity gives the mass–energy content of a system. The SI unit of mass is the kilogram (kg).

Mass is not the same as weight, even though we often calculate an object's mass by measuring its weight with a spring scale instead of comparing it to known masses. An object on the Moon would weigh less than it would on Earth because of the lower gravity, but it would still have the same mass.

For everyday objects and energies well-described by Newtonian physics, mass describes the amount of matter in an object. However, at very high speeds or for subatomic particles, special relativity shows that energy is an additional source of mass. Thus, any stationary body having mass has an equivalent amount of energy, and all forms of energy resist acceleration by a force and have gravitational attraction.