In mechanics we consider that there are three elemental quantities called dimensions: mass, length and time. All physical quantities that we deal with can be expressed in terms of these, which will be abbreviated M, L, and T. (Not to be confused with the BLT, available at Denny's)

1. What are the dimensions of velocity?

velocity = length/time velocity = L/T

2. Express density in terms of its basic dimensions.

density = mass/volume M/L3

3. Write out the dimensions of (a) acceleration and (b) weight.

a) a = L/T2

b) W = mg W = ML/T2

4. Write out the units of (a) acceleration and (b) force in the English system.

a) a = L/T2 a = ft/sec2

b) F = ML/T2 F = (slug * ft)/sec2

Let's talk briefly about units. There are two systems of units commonly used by physicists the SI or International System which uses meters, kilograms and seconds, and the cgs system which uses centimeters, grams and seconds. While the International System is being pushed as THE world system, cgs is still popular and a third system, the English system is used in the United States for some engineering and manufacturing.

In this text I've used the SI most, particularly toward the end of the book. I interject the other two systems to familiarize you with one of life's hard realities and to force your awareness of units as you solve problems. In the end you will have a better mastery of SI than you would if all problems had used those units just as you have a better understanding of English after you've learned a second or third language.

5. To which system do each of the following units belong?
a) kilogram
b) foot
c) newton
d) dyne
e) centimeter
f) second

a) SI

b) English

c) SI

d) cgs

e) cgs

f) all

6. Write the units of pressure and density in the cgs system.

P = force/area P = dyne/cm3 P = [(gm cm)/sec2]/cm2 P = gm/(cm sec2) p = mass/volume p = gm/cm3

I hope you have noticed the different use of units and dimensions. Dimension refers to the type of measurement, such as length. Unit refers to the particular standard, such as centimeter, foot, meter, or Angstrom.

7. Think up a meaningful physical quantity for each of the following and state what unit system it is in.
a) nt/m²
b) gm/cm
c) ft³/sec
d) cm³/sec²

a) Force per area (pressure), SI units

b) Mass per length (as in wire), cgs units

c) Volume per time (as in fluid flown through a pipe), English units

d) Volume per time per time (rate of change of fluid flow), cgs units

8. If the SI units of a newton are kg-m/sec², what are the basic units of the following?
a) nt²
b) nt-sec
c) kg/nt
d) (nt²-sec)/kg⁵

a) [(kg-m)/sec2]2 (kg2m2)/sec4

b) [(kg-m)/sec2](sec) (kg-m)/sec

c) kg/[(kg-m)/sec2] sec2/m

d) [[(kg-m)/sec2]2-sec]/kg5 m2/(sec3kg3)

9. As an object moves through a fluid the drag force on it can be expressed as F = -kv² where v is the velocity and k is some constant. (The negative sign indicates that the force acts oppositely to the direction of motion.)
a) What would be convenient units for k in the SI system?
b) What are the basic dimensions of k?

a) nt = k(m/sec)2 k = nt/(m/sec)2 k = (nt-sec2)/m2

b) Basic dimensions are: (nt-sec2)/m2 [(kg-m)/sec2] * [sec2/m2] kg/m M/L

10. Write the unit that corresponds to the one given, but in the system stated.
a) kg in English
b) pound in cgs
c) nt-m in cgs
d) dyne/cm² in SI
e) ft/sec² in SI
f) slug²/nt-m in cgs
g) gm-cm²/dyne-sec³ in SI

a) slug

b) dyne

c) dyne-cm

d) nt/m2

e) m/sec2

f) gm2/(dyne-cm)

g) (kg-m2)/(nt-sec3)