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Physical Quantities, Units and Dimensions
A physical quantity is anything which can be measured. Examples include: pressure, area, time, voltage, frequency. To measure a physical, quantity a standard which is called a unit is selected and all other measurements of that quantity are compared to the standard unit. This standard of measurement is called The International System of Units (S.I.).
Table 1.1 Fundamental Quantities
Base Quantity Abbreviation of Name of Unit Dimension Unit Length m metre L Mass kg kilogram M Time s second T Electric Current A ampere I Temperature K kelvin Ѳ Amount of substance mol mole N Luminous Intensity cd candela J
S.I. Units comprise of any combination of these base or fundamental units. For example: m/s or kg/m3. S.I. Units are also given special names, for example the S.I. Unit of Force is called Newton (N). All other quantities must be combinations of base or fundamental quantities given in Table 1.1 above and are called derived quantities. Some common examples of derived quantities and their units are shown in Table 1.2 below.
Table 1.2 Derived Quantities and Corresponding Units
Derived Quantity Unit Name Unit Symbol Velocity metre per second ms-1 Acceleration metre per second2 ms-2 Force Newton N = kgm s-2 Momentum Newton second Ns = kgm s-1 Pressure Pascal Pa = kgm-1s-2 Energy Joule Nm = kgm2s-2 Power Watt W = kgm2s-3 Volume metre3 m3 Frequency Hertz Hz = s-1 Charge Coulomb C = As Electromotive Force Volt V = kgm2s-3A-1 Resistance Ohm Ω = kgm2s-3A-2 Capacitance Farad F = kg-1m-2s4A2 Magnetic Flux Weber Wb = kgm2s-2A-1 Magnetic Flux Density Tesla T = kgs-2A-1
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Example 1.1 To determine Fundamental Quantities from Derived Quantities
What are the fundamental quantities of speed?
� � = �
� ℎ Therefore the dimensions = � of speed are length and time
Problems 1.1
a. What are the fundamental quantities of density? b. What are the base quantities of acceleration? c. What are the fundamental quantities of force? d. What are the fundamental quantities of momentum?
Example 1.2 To determine Base Units of Derived Quantities
What are the base units of voltage?
� � = � � �
� � = × � � = �
= × � ℎ = × � � � � = � therefore − � = − � = − = − = hence ℎ = � − � − − = � = �
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Problems 1.2
a. Determine the fundamental quantities and the base units of power.
b. Determine the base units of specific heat capacity
c. Express the derived units for pressure (Pa) in terms of base units.
Dimensions
All seven (7) base quantities used in SI have their own dimensions which are shown in Table 1.1 above. The dimensions of derived physical quantities is the way it is related to the dimensions of the base quantity. To find the dimensions of a derived quantity an equation involving the quantity must be used.
We use the square brackets [] to represent the statement “the dimensions of”
Example 1.3 To determine the Dimensions of Derived Quantities
a. Find the dimensions of power.
� � � � = � − [ ] = � therefore [ � ] = � − [ ] b. If the Earth is taken to be a =uniform � sphere or radius r and density ρ, the gravitational intensity at the surface is given by , where k is a constant. What are the dimensions and the base units of k? = �
= − [ ] = � [ ] = − [ ] =
− � − [ ] = therefore and base units are − − − − [ ] = � D. Whitehall 4
Problem 1.3
a. Find the dimensions of momentum.
b. Find the dimensions of pressure.
c. Find the dimensions of the universal gravitational constant G, given that the gravitational force between the two point masses is given by the equation =
d. When a gas flows through a small diameter its behavior is governed by the equation
− = √ � where k a dimensionless constant r the radius of the tube , the pressures at the end of the tube l the length of the tube M the molar mass of the gas (kgmol-1) R the molar gas constant (JK-1mol-1) T the temperature
Use the equation above to find the dimensions and the base units of
e. The equation of state for one mole of a real gas is
+ − = � where P pressure V volume T temperature
Determine the dimensions and the base units of a, b and R
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Dimensional Homogeneity of Equations
An equation is homogeneous or dimensionally correct when the equation is balanced on both sides in respect to the units of the quantities in the equation.
Example 1.4 To determine the Dimensions Homogeneity of Equations
a. Show that the equation , is dimensionally homogeneous. LHS: = +
RHS: [ ] = − [ ] = � � = − Since the dimensions[ of each] = �term on� RHS= of the equation are equal to the LHS of the equation, the equation is homogeneous.
Problem 1.4
a. Show that the equation is dimensionally homogeneous. b. Use dimensional homogeneity� = u to +determine a� which of the following equations is correct
� � = √� � = √ c. Check whether the following equations are dimensionally correct
i. where m is mass, F is force and T is time � � = � ii. where c is speed, T is tension and µ is mass per unit length �
= õ
d. What is the base units of R that would make the equation homogeneous, where P is pressure, V is volume, n is the number of moles, T is temperature? � = � e. Show that the equation is dimensionally consistent and state the SI units of k. Given that P is pressure, h is height, ρ is density, g is gravitational force, v is velocity and k is constant. + ℎ + ½ =
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Dimensional Analysis
Since equations are dimensionally homogeneous, we are able to make predictions about the way physical quantities relate to each other.
Example 1.5 Dimensional Analysis
The period of a pendulum can be calculated by using the formula
where � k is a dimensionless = constant, m is mass, l is length, g is gravity x, y and z are unknown indices.
Determine the values of x, y and z. State the final relationship.
[ � ] = [ ][ ][ ] − � = � + − Equating the LHS of �the = equation to� the RHS of the equation we get:
(for M)
(for L) =
(for T) = +
When we solve these three equations we get = that − , , The relationship is therefore = = − = ½ −½ � = Hence � = √�
Note: We cannot determine the value of the dimensionless constant k by using Dimensional Analysis.
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