0.15 Phy1120: force -- center of gravity  (Page 2/8)

Supplemental material

I recommend that you also study the other lessons in my extensive collection of online programming tutorials. You will find a consolidated index at www.DickBaldwin.com .

General background information

The gravitational constant

The gravitational constant, G, is a constant that is used in the calculation of the gravitational attraction between objects with mass. The value of G is approximately:

G = 6.67 * 10^(-11) *m^3*kg^(-1)*s^(-2), or

G = 6.67 * 10^(-11) N*(m/kg)^2

According to the law of universal gravitation, the attractive force (F) between two bodies is proportional to the product of their masses (m1 and m2), and inversely proportional to the square of the distance (r) between them:

F = G*m1*m2/r^2

Newton's law of gravitation

Newton's law of gravitation proposes that every small element of matter, m, attracts every other small element of matter, m' with a force that isproportional to the product of the masses and inversely proportional to the square of the distance between them, so that at the small element level,

F = G*m*m'/r^2

As a result, we tend to fall toward the earth unless we are supported by upward forces, such as the floor that we are standing on.

An important simplification

It would be very difficult to deal with the attraction of gravity if we were required to deal with every object (including the earth) in terms of the manysmall elements of mass that make up the object.

Newton simplified that process by proving mathematically that the earth regarded as a sphere attracts abody as if the whole mass of the earth were concentrated at its center.

Almost parallel forces

The attraction of the earth on the elements of our body produces forces on thoseelements that are almost parallel. Those forces lie along lines that all end at a point that is almost 4000 miles beneath the surface of the earth.

At that distance, each pair of force lines forms the sides of a triangle with extremely long sides, an extremely small angle, and an extremely narrow base. Asa practical matter, therefore, we can assume that those forces are parallel insofar as estimating the effectthat the sum of all of those elemental forces have on our bodies.

The center of gravity C.G.

It can be proved that:

The resultant of the weights of all the elements of a rigid body passes through a certain fixed point commonly called its center of gravity, C.G., regardlessof the orientation of that body relative to the earth.

For purposes of computations involving statics, the mass of a rigid body may be considered as if concentrated at it C.G., and its weight may be considered to centerat this same point.

(Recall that the weight of an object is the force required to cause the mass of that object to fall toward the center of the earth with an acceleration ofapproximately 32.2 ft/s^2 or 9.81 m/s^2.)

A plumb-bob or plummet

A plumb-bob or a plummet is a weight, usually with a pointed tip on the bottom, that is suspended from a string and used as a vertical reference line, or plumb-line.