Some Basic Units of Measurement in Mechanics

Laboratory Exercise No. 1

"Laboratory Manual for General Physics", Bernard D.Kern. Part One: Mechanics, Heat and Sound, Revised Printing, 1996

Copyright C 1968, 1971, 1996 by Bernard Donald Kern

DISCUSSION

The procedures which we shall use in Mechanics can be reduced to measurements of length, mass, and time. For the time being, we can consider mass as "quantity of matter". The Metric and English units for these are

	Metric	Abbreviation	English	Abbreviation
length	meter	m	foot	ft
mass	kilogram	kg	slug	slug
time	second	s	second	s

The Metric units in this list (plus four others involving heat and light) are usually called the "MKS" system of units. The size of each unit has been arbitrarily (but logically) defined; international standards exist. The equations which we shall use will be given in these units, but according to custom and convenience we often shall make readings in fractions or multiples of these units in order to match the size of the thing being measured. The familiar English system of units shall on occasion be used in order to develop a better intuitive feeling for the size of things.

In order to measure a physical quantity, one must choose a standard of reference. In order for you to measure a length, you must have selected some length as a standard. To make a measurement of length the procedure is to compare that length with your standard length. The result will be a number and a unit which is the name of the standard; e. g., 1.65 meters.

Consider the measurement of a length both in meters and in inches. The physical concept is that the length of the object is unchanged regardless of the chosen system of units in which that length is given. Thus the measured length (number and a unit) describes the same thing in both systems of units. Suppose that you measure a length as 45.00 centimeters and alternatively as 17.72 inches; then

(1)

45.00 centimeters = 17.72 inches (mixed units of length)

Dividing both sides first by 45, and then by 17.72, gives

(2)

1 centimeter = 0.394 inches

1 inch = 2.54 centimeters

Useful "conversion factors" are obtained by dividing by the unit which appears on the left-hand side, giving

(3)

1 = 0.394 inches/centimeter

1 = 2.54 centimeters/inch

The process of weighing an object also is the making of a comparison with a standard. "Weight" is defined as the downward force on an object caused by gravity; weight is equal to the mass times the acceleration of gravity, g. So if we compare weights we are also comparing masses. In the laboratory, we compare masses in the MKS system. Using a balance, we observe that the gravitational forces which act on two masses are equal, and conclude that the masses are equal. If an unknown mass is measured in two sets of units, we can use equations which are similar to Eqns. 1 to 3 to obtain the ratio of units.

There is a custom in the English system of stating the amount of material in an object as weight in pounds (a unit of force) instead of in slugs (a unit of mass). The conversion factor equation is best remembered as mg = weight,

(4)

(1.0 slug)g = weight (mixed mass units)

(1.0 slug) (32.174 ft/s2) = 32.174 pounds

I.e., 1.0 slug (mass) is equivalent to 32.174 pounds (weight).

Angular measurements in radians can be reduced to measurements of length. One radian is that angle which is subtended by the arc of a circle with length equal to the radius of the circle. For example, assume numerical values for an arc length of 0.45 meters on the circumference of a circle with radius of 1.5 meters. The number of radians is

(5)

arc length/radius = 0.45 meters/1.5 meters = 0.30 radian

Note that the physical dimension, the meter, has cancelled. Hence a radian has no physical dimension of length, mass, or time, and is said to be dimensionless.

A less-familiar angular quantity is the steradian, the unit of solid angle. Imagine a cone with its apex at the center of a sphere of radius R; this cone intersects the surface of the sphere in a circle and inside of this circle there is an area, A, on the surface of the sphere. The solid angle is at the apex of the cone, enclosed by the surface of the cone. It is, in steradians, equal to the area A of the spherical surface divided by the square of the radius R of the sphere, thus

(6)

No. of steradians = A/R2

One steradian is that solid angle which is subtended by one unit of area on the surface of a sphere of unit radius. The steradian is a dimensionless quantity. For small solid angles, it is usually sufficiently accurate to use a plane area for A as an approximate substitution for the spherical surface area described above.

PROCEDURES

Length: Measure the three dimensions (length, width, and depth) of the wooden block in centimeters and in inches. The extreme end of the meter stick is probably worn and should not be used; rather, use the 1 cm or 10 cm mark as a starting point.
Mass: The calibrated mass is marked in English units. Use the trip balance to determine its mass in kilograms. On your data sheet, list each of the individual standard masses ("weights") and do the sum on paper to reduce the possibility of making an error.
Radian: Measure the diameter of the circular disk in centimeters; make a total of four determinations along different diameters and take the average of the four. Measure the length of the circumference of the disk with the string, in centimeters.
Steradian: Place the disk flat on the table with the end of the meter stick, held vertically, resting at its center. The solid angle inside of the imaginary cone which is subtended by the circular disk and which has its apex at the top of the stick is to be measured. Note as data the length of the stick and the diameter of the disk.

CALCULATIONS

For each of the measurements of the block, calculate the ratio of the units in the two systems. Take the average of the three as your best value. Compare your ratio with that in Eqn. 3.
Calculate the ratio of the English mass unit to the MKS mass unit and the reciprocal, the ratio of the MKS value to the English system value. Compare your ratios with those in Eqn. 4.
Use Eqn. 5 to determine the number of degrees per radian, and the reciprocal value. Compare your result with the expected 2*PI radians per 360 degrees.Use Eqn. 6 to calculate the number of steradians in the solid angle which you have constructed. Prove that the surface of any sphere subtends 4*PI steradians at the center of the sphere

The Java Applet below was created, designed and compiled by Vladimir Sorokin, 1997. See Copyright message and Acknowlegment message

CHECK THE CORRECTENESS OF THE RESULTS YOU RECEIVED IN CLASS:

If you are using Java aware browser, such as for example Netscape 3.0, then you could use "Physics Unit Converter" Java applet below to convert various kinds of Physics units into one another. Select the type of the units in the selection box on the top and chose a couple of units that you would like to do conversion between. Then, if you enter a numerical value in one of the text fields, you get the converted value in the other field.

The numerical value in the right textfield is recalculated every time you:

double click on a new unit (either on the left or on the right);
select a new type of units ("Length", "Mass", ... );
edit the numerical value on the left and press enter

The numerical value on the left is recalculated only if you edit the numerical value on the right and press enter.

You can also use this applet to reference major Physics constants in the bottom part of the applet.

The source of information on conversion and constants used in the creation of this applet is shown in the reference window ( press the button "Reference" ).

If your browser recognized the applet tag, you would see an applet here.

Distribution files

Comments/Suggestions/Questions: vsoro00@pop.uky.edu

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