1. Secondary Quantities Notation Dimensional Homogeneity
EGR 101
Secondary Quantities
Notation
Dimensional Homogeneity
EGR 101

2. More on Secondary Quantities
Important secondary quantities: Force: Energy (work): Power:
Force (N): Newton (must be capital); .1N = 1kg m/s2
Work (N.m): Joule; (Joule must be capital). 1J = 1kg m2/s2 potential energy ½ m v2
All Energies (J)
Power (J/s): Watt;
Used the example of two students consuming identical hamburgers with the same amount of energy, with one student consuming the hamburger in 5 minutes and the second student consuming the hamburger in 30 minutes. Other examples included running to burn energy over different time interval, and consuming electrical energy over different time intervals. Thus we defined the combination of J/s as W (Watt).
EGR 101

3. Questions: What is a kilowatt-hour? Where is it used? EGR 101
1kw-hr = 1000 W/kW *3600 s/hr = 1000*3600 W-s *1J/s /W = 3.6M J
EGR 101

4. In-Class Activity
A study of households and businesses in the Boston, Massachusetts area found that air conditioning units were used for an average of 4600 hours per year. Determine the total annual cost of electricity required to operate a 10,000 Btu/hr air conditioning unit in the Boston area if the electric rate is $ 0.071/kWhr.
10,000 BTU/hr = 1x10^4BTU/hr*2.93x10-4kW/(BTU/hr) = 2.93kW
2.93kW * 4600 hrs/yr = 13,478 kWhrs/yr
13,478 kWhrs/yr*$0.071/kWhr = $956.94/yr
Pout/Pin = eff; ideally eff = 1 but in reality it’s always less than 1.
Allow about 10 minutes for this activity
Input power = .25/.8 =.312 hp .312 hp * W/hp = W
Total energy = power * time E = W * 4 hrs/wk * 52 wk/yr * 1 yr = 48,470.5 Whr or kWhr kWhr * 2.665x106 J/kWhr = MJ kWhr * 3415 Btu/kWhr = 165,525 Btu
EGR 101

5. Notation
The units of variables will be referred to by putting the variables in brackets Consider the equation: v-vo = a t
What is the equation saying?
EGR 101

6. [v] will refer to the dimension of v [t] will refer to the dimension of t
In order to determine the units of an unknown (say a), we need to be able to write equations in terms of units, e.g.
EGR 101

7. Rules of Homogeneity
Definition: An equation is said to be dimensionally homogeneous if all terms separated by plus and minus and equal sign have the same dimension.
Consider the previous example:
In order to be homogeneous,
EGR 101

8. Rule 1
If the dimensional quantities are replaced by their primary units the equation should reduce to an identity.
In our example, what are the primary units?
Good place to bring up ways to write accel = m/s/s
EGR 101

9. Rule 2
Dimensions do NOT add or subtract. In order to add or subtract variables, they must have the SAME units.
In our example, [v] = [vo]. If you subtract them you have no units on the left
EGR 101

10. Rule 3
Dimension CAN multiply and divide.
EGR 101

11. In-Class Activity If P1 = 400 W and P2 = 12 Btu/minute, what is P1+P2?
1BTU/hr = 2.93x10-4 kW = .293 W
P2 = 12 BTU/min * 60 min/hr *.293 W/(BTU/hr) = 720 * .293 W = W
P1 + P2 = 400 W W = W
EGR 101

12. Exercises
A relationship between Force F in (N), distance x in (m), mass M in (kg) and speed v in (m/s) is suggested as Is this equation dimensionally homogeneous?
Yes
EGR 101

13. A relationship between Force F in (N), time t in (s), mass M in (kg) and speed v in (m/s) is suggested as: Is this equation dimensionally homogeneous? No
EGR 101

14. Prefixes
As you’ve already seen, we’ll deal with both very large numbers and very small numbers.
We will use scientific notation and/or engineering prefixes to represent these numbers
EGR 101

15. Examples
2x103 Volts =
A =
1.3x10-6 C =
10x107 Hz =
EGR 101

16. Some Quantities Must Be Dimensionless
In Calculus you will see eax
ax must be dimensionless (no units) 1/1
Example, if x is in m, a is in 1/m
If x is in s, a is in 1/s
Some dimensionless quantities need units to make sense
Example if v = vo e-at v and vo have units volts
EGR 101

17. Class Activity
In an electrical circuit the current i(t) in A (Amperes) changes with time t in s according to the function i(t) = e-2t.
a) How could this function be dimensionally consistent knowing that the exponential function is always dimensionless (RHS), and that i(t) is in A on the LHS?
b) What are the units of the constant 2 in the exponential function?
Implied 1 A multiplies e-2t
The unit of the constant is 1/s
EGR 101

18. More on Units Remember x = cos(θ)? What are the units of θ? EGR 101
Θ is in radians.
Draw circle on board and demonstrate relation between radius and radians
EGR 101

19. What are the units of ω in y(t) = cos(ωt) if t has units s?
ω is called the angular frequency
EGR 101