Mass. ) or the slug (lb f s²/ft)

Transcription

1 MASS VS. FORCE

2 Mass As stated elsewhere, statics is the study of forces on a stationary body. To this end, we must understand the difference between mass and force and how to calculate force if given the mass of a body. How this is done is a bit different between working in the inch-pound system vs. working in the S.I. system Mass describes a quantity of material, not the weight (force) of the material. In the S.I. system, it is common to measure this in terms of grams or kilograms. In the inch-pound system, the typical unit of mass is the pound-mass (lb m ) or the slug (lb f s²/ft) 2

3 Force Force is defined as an action that cases some form of reaction. In the S.I. system, force is measured in Newtons (kg m /s²). In the inch-pound system, force is typically measured in pounds-force (lb f ). For large forces, we tend to use the units of kips (1000 lb f ). Of course, we do have the units of short ton (2000 lb f ), long ton (2240 lb f ) and metric ton (2205 lb f ). However, these measurements are used primarily for cargo transport, not for statics calculations 3

4 Newton's Law Newton's law states force is the product of mass and acceleration. In statics, we are only concerned with acceleration due to gravity. As such, Newton's law can be written as F = m x g In the SI system: F = Newtons m = kilograms g = 9.81 m/s² For example, to determine the force of a body with a mass of 1875 kg, we would write: F = 1875 kg x 9.81 m/ s 2 = 18,393.7 kg m/ s 2 = 18,400 N (Expressed to three significant digits) 4

5 Newton's Law In the inch-pound system: F = lb f m = slugs g = ft/s 2 The slug is an old unit of mass. It is a fundamental unit, but can be considered a lb f s²/ft. This unit is still used in some engineering and engineering technology disciplines today and was used extensively in the past. In fact, it is critical to understand Newton's law was written using slugs as the unit of mass, not the lb m 5

6 Newton's Law For example, to determine the force exerted (weight) by a mass of 5.0 slugs, we would write: F = 5.0 slugs x ft / s 2 = slugs ft s 2 = 160 lb f Using proper number of significant figures A more common unit of mass in the inch-pound system is the lb m. However, the unit of lb m cannot be used in Newton's law as written 6

7 Newton's Law Redux To do so, we must modify Newton's law as follows: F = m x g / g c where: F = lb f m = lb m g = ft/s² (acceleration due to gravity) g c = (ft lb m )/(s² lb f ) (gravitational constant) Note the acceleration due to gravity and the gravitational constant have the same numeric value, but are very different quantities. The acceleration due to gravity is considered constant for all of our calculations, but, in fact, it is not 7

8 Newton's Law Redux The acceleration due to gravity varies across the face of the earth due to the uneven distribution of mass around the globe and to the fact the earth is an oblate spheroid (pumpkin shaped). This results in the distance from the center of the earth to the equator being approximately 13 miles greater than that to the poles. Fortunately, none of this is of concern to us when determining the weight of a mass 8

9 Newton's Law Redux On the other hand, the gravitational constant is a constant. Its value does not vary. One should note, in the S.I. system, g c is unitless and has a value of one If we do a unit analysis, we find: F = lb m x ft / s 2 ft lb m lb f s 2 = lb f Note that 'g' and 'g c ' divide out to equal one (numerically). In the end, the numeric value of mass and the numeric value of force are equal when mass is expressed in lb m 9

10 Newton's Law Redux For example, if your body has a mass of 185 lb m, then it has a weight of: F = 185 lb m x ft / s ft lb m s 2 lb f = 185 lb f This often generates the question If lb m and lb f are numerically equal, why bother? The answer is twofold. First, mass and force define two very different quantities, and second, they are not always numerically equal. Let's look at some examples. 10

11 Example 1 The mass of a body does not change, but the force exerted by the body due to acceleration does. For example, a body with a mass of 5 slugs, weighs 160 lb f. However, if placed on an elevator accelerating upward at 5 ft/s², the weight (the force exerted on the floor of the elevator) changes and can be recalculated as: F = 5.0 slugs x ft / s ft / s 2 = 5.0 slugs x 37.2 ft / s 2 = 186 slug ft = 186 lb s 2 f Although the weight of the body did change (albeit, temporarily), the mass of the body did not 11

12 Example 2 In a similar fashion, assume a body with a mass of 160 lb m. If this mass is placed in an elevator accelerating downward at 5 ft/s², the force it exerts on the floor of the elevator can be calculated as: F = 5.0 slugs x ft / s ft / s 2 = 5.0 slugs x 27.2 ft / s 2 = 136 slug ft = 136 lb s 2 f Again, the weight of the body has changed while the mass remains constant 12

13 Mass vs. Force This may seem rather academic to many. It is not. When you are dealing with skyscrapers or suspension bridges that sway in the wind, when you deal with belt drives, chain drives or clutch systems on machines, when you deal with anything in motion, this is a particularly important concept Fortunately, statics deal with objects at rest. The force caused by an object will always be that of acceleration due to gravity only. External accelerations will be dealt with in a dynamics course 13