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Its volume would be multiplied by the cube of 2 and become 8 m 3. The original cube (1 m sides) has a surface area to volume ratio of 6:1. The larger (2 m sides) cube has a surface area to volume ratio of (24/8) 3:1. As the dimensions increase, the volume will continue to grow faster than the surface area. Thus the square–cube law.
In classical mechanics a physical body is collection of matter having properties including mass, velocity, momentum and energy. The matter exists in a volume of three-dimensional space. This space is its extension. Interactions between objects are partly described by orientation and external shape.
Newton's laws are often stated in terms of point or particle masses, that is, bodies whose volume is negligible. This is a reasonable approximation for real bodies when the motion of internal parts can be neglected, and when the separation between bodies is much larger than the size of each.
To make this into an equal-sided formula or equation, there needed to be a multiplying factor or constant that would give the correct force of gravity no matter the value of the masses or distance between them (the gravitational constant). Newton would need an accurate measure of this constant to prove his inverse-square law.
The experimental determination of a body's center of mass makes use of gravity forces on the body and is based on the fact that the center of mass is the same as the center of gravity in the parallel gravity field near the earth's surface. The center of mass of a body with an axis of symmetry and constant density must lie on this axis.
The surface-area-to-volume ratio has physical dimension inverse length (L −1) and is therefore expressed in units of inverse metre (m-1) or its prefixed unit multiples and submultiples. As an example, a cube with sides of length 1 cm will have a surface area of 6 cm 2 and a volume of 1 cm 3. The surface to volume ratio for this cube is thus
A body force is simply a type of force, and so it has the same dimensions as force, [M][L][T] −2. However, it is often convenient to talk about a body force in terms of either the force per unit volume or the force per unit mass. If the force per unit volume is of interest, it is referred to as the force density throughout the system.
If a first body of mass m A is placed at a distance r (center of mass to center of mass) from a second body of mass m B, each body is subject to an attractive force F g = Gm A m B /r 2, where G = 6.67 × 10 −11 N⋅kg −2 ⋅m 2 is the "universal gravitational constant". This is sometimes referred to as gravitational mass.