CHAPTER 1 : PHYSICAL QUANTITIES AND UNIT
1.1 Base Quantities and SI Units
Physical Quantity
1. A physical quantity is a quantity that can be measored.
2. Eg: Length,time,weigh,velocity, and momentum.
3. A physical quantity can be expressed in numerical magnitude together with a unit.
4. Eg : Mass of a book = 520.3 g
(Physical quantity) (Numerical magnitude) (Unit)
5. Physical quantities can be classified as base quantities or derived quantities.
5. Eg of base quantities : 
 length, mass, time are stand alone quantities that cannot be derived from other
quantities.
6. Eg of physical quantities: 
 volume, velocity and acceleration that can be derived from other quantities.
Base Quantities and SI units
Base quantity (Symbol) SI unit (Symbol)
Length (l) Meter(m)
Mass (m) Kilogram(kg)
Time (t) Second(s)
Electric current (I) Ampere(A)
Thermodynamic temperature (T) Kelvin(K)
Amount of substance (n) Mole(mol)
Eg: Derived Quantities and Derived Units
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Derived quantity Derived unit Abbreviation

Area (A) m2 
Volume (V) m3 
Speed, velocity (v) ms1 
Acceleration (a) 
Density(ƿ) 
Current density (J)
Electric charge (Q) Coulomb (C)
Frequency (f) Hertz (Hz)
Force (F) Newton (N)
Pressure, stress (p) Pascal (Pa)
Energy (E) Joule (J)
Power (P) Watt (W)
Electric potential difference (V) Volt (V)
Capacitance (C) Farad (F)
Electric resistance (R) Ohm (Ω)
Magnetic flux (ϕ) Weber(Wb)
Tesla (T)
Magnetic flux density (B) Hendry (H)
Inductance (L)
Eg :1
2
Gravitational force is given by F = GM1M2
R²
where M1 and M2 are the masses of two objects whereas R is the separation between the two
objects. Find the derived units of G, the universal gravitational constant. (m³ s² kg¹)
Prefixes Name Symbol
Factor
10²4
10²¹
10¹8
10¹5
10¹²
109
106
10³
10²
10¹
101
102
103
106
109
1012
3
1015
1018
1021
1024
1.2 Dimensions of Physical Quantities
Dimensions of base Quantities
1. A physical quantity can be expressed in ___________________ expression.
2. Some quantities do not have ___________ and are thus dimensionless.
3. Dimensions of base quantities are given in the table.
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Base quantity Dimension
Mass
Length
Time
Electric current
Temperature
Amount of substance
4. The notation of dimension for a physical quantity is expressed as
_________________________. For instance, the dimension of displacement, s, can
be expressed as [s] = _________.
To Determine the Dimension of a Derived Quantity
1. Dimensions of derived quantities can be obtained from the formulae of the physical
quantities. For example, dimensions of
(a) density
(b) momentum
2. Dimensions of a physical quantity can be obtained by comparing its derived units. For
instance, the derived units of accelerations is ms², hence
Example of dimensions of derived quantities, determined by comparing the corresponding
derived units:
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Derived quantity Relationship with other quantities Units Dimensions
Force mass x acceleration
Pressure
Force
Energy area
Force x displacement
Eg:
1.The moment of inertia I, of a uniform rod of mass m and length l about an axis perpendicular
to one end of the rod is given by I = 1/3 ml². Find
(a) the units of I and
(b) the dimensions of I.
Checking the Homogeneity of an Equation
1. Examining the dimensions of physical quantities is useful because it enables us to check
the validity of a given equation. For a valid equation, it is dimensionally homogeneous,
e.g quantities on both sides of an equation must have the same dimensions.
2. Only quantities with the same dimensions can be _________ of _______________. For
example, an equation describing the motion of an accelerating body in terms of its final
velocity, v, initial velocity, u, acceleration, a and time, t, is wrongly given as
v = ut + at
Examine the dimensions of each term:
The term on the left side of the equation does not have the same dimensions as the left
on the right hand side; therefore, ___________________________________________.
Eg :
1. Under uniform acceleration, motion of an object with velocity, v, is represented by v² = a + bx
where a and b are constants and x is a variable for displacement . If both a and b have
dimensions, find the dimensions of
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(a) a
(b) bx
(c) b
Dimensional analysis
1. By means of dimensional analysis, we can derive an expressions to relate a physical
quantity with other physical quantities.
2. The value of any dimensionless constant involved in the empirical equation such as 2π
cannot be determined by this method but can be determined experimentally.
3. Note that a dimensionless constant _____________________________.
Eg. 1
In simple pendulum experiment, a student makes an assumption that the period of oscillation of
the pendulum, T, is related to the mass, m, of the pendulum bob, the length, l, of the string and
also the acceleration due to gravity, g. Derive an equation for T by dimensional analysis.
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1.3 Scalars and Vectors
1. A _____________________ is a physical quantity which has _______________ only.
Eg: of scalar quantities are ___________, _____________and _____________.
2. A ______________________ is a physical quantity which has both _______________
and __________________. Eg; _________________, _________________ and
_____________________.
P
A vector is normally represented by a bold print letter such as P or P and its magnitude by [P]
or p. The length of the arrow represents its magnitude whereas the direction of the arrow
represents the direction of the vector.
4. If vectors P and Q have the same magnitude and direction even at different locations as
shown below, 3N 3 N
PQ
Then P = Q
Eg: 1
State the scalar quantities from the following physical quantities.
Velocity, Acceleration, Electric potential, Electric field strength and Work done.
Addition of Vectors
1. The addition of two vectors, P and Q that is the resultant vector, R = P + Q can be
represented by a triangle of vectors,
R
Q
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P
2. Notice that the resultant vector R, joins the beginning point of P(1ˢt vector) and the end point
of Q (2nd vector).
Subtraction of vectors
1. A negative vector –Q is a vector Q acting in the opposite direction.
2. Figure below shows the subtraction of two vectors, P – Q, that is represented by a
triangle of vectors.
PQ
Q Q
This is equivalent to the addition of a positive vector and a negative vector
R = P – Q = P + (Q)
Scalar Product
1. The scalar product (dot product) of vector A and vector B is defined as
where ϴ is the angle between A and B.
2. The resultant value of a scalar product is a ____________________________.
3. For instance, work done is defined as the scalar product of force and displacement in the
direction of force, W = F ● S = Fs cos ϴ, where ϴ is the angle between F and s.
4. A ● A = A² since the angle between the two vectors is zero.
5. A ● B = 0 if the angle between the two vectors is 90º.
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Vector Product
1. The vector product (or cross product) of vector A and vector B is defined as
where ϴ is the angle between A and B.
2. The resultant value of a vector product is ______________________.
3. For instance, torque is defined as the vector product of distance r from the axis of rotation
and force F, = r x F= rF sin ϴ where ϴ is the angle between r and F.
4. The direction of a torque is ___________________ to the plane formed by r and F, pointing
in the direction according to the ______________________________.
5. From the definition of vector product, we conclude that
This means that B x A points in the opposite direction to A x B with the same magnitude.
6.
A and B form a parallelogram with an area that equals AB sin ϴ
Example:
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Figure shows vector A and vector B that are placed on a horizontal plane. The crossproduct of
the vector , Ax B, is equal to vector C pointing in the direction as shown. If A = 25 m, B = 40 N
and the angle between A and B is 30º, find
a) The magnitude of C
b) Magnitude and direction of B x A
(500 Nm, 500 Nm opposite direction to C)
Resolution of Vectors
1. A number of vectors can be resolved into two vectors, ________________ and
_______________, so that a resultant vector can be calculated.
2. Figure below shows three vectors, A, B and C, acting in three different directions.
Resolving horizontally,
Resolving vertically,
Resultant vector,
the direction of the resultant vector,
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Example: 1
Figure above shows three forces 20 N, 40 N and 50 N acting at a point O. Calculate
(a) The magnitude and
(b) The direction of the resultant force.
(60.087 N, 3.1815 N, 60.171 N, 3.0º to the horizontal)
2.
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6.
1.4 Uncertainties in Measurements
Errors in Measurement
1. An error in measurement is the different between a _________________ and the actual
________________________.
2. All measured values have ______________. The accuracy of a measured value depends on
the sensitivity of the measuring that measures to the nearest decimal places.
3. The length of a wire measured by using a ruler is recorded as 34.2 cm. It is incorrect to
record the value as 34.20 cm because the ruler is not reliable enough to measure to the nearest
0.01 cm. Similarly, it is inappropriate to record the value as 34 cm because the ruler is sensitive
enough to measure with accuracy until 0.1 cm.
4. If a vernier calipers, which is more sensitive is used, the recorded value of 34.20 cm is
acceptable.
5. Hence, the measured value by using a vernier calipers is more _____________ than by using
a ruler. This is because a ruler can measure values with accuracy until _____________ but a
vernier calipers can measure values with accuracy until _________________.
6. Furthermore, measurement made by using a micrometer screw gauge is more accurate than
by using a vernier calipers because it can measure with accuracy until ___________________.
7. 0.1 cm, 0.01 cm and 0.001 cm represent the ________________________ for a ruler, a
vernier calipers and a micrometer screw gauge respectively.
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Systematic Error
1. Systematic error is due to constant deviation of the readings in one direction only, that is
either consistently too high (always positive) or consistently too low (always negative) from its
true value.
2. Hence, using statistical method or taking the mean value of repeated measurements does not
help to reduce systematic errors.
3. Some examples of _____________________________:
(a) The ______________________ of a measuring instrument such as vernier calipers or
micrometer screw gauge.
(b) ________________________________ of a measuring instrument.
(c) _________________________ such as human reaction in the measurement of time by
using a stopwatch.
(d) __________________________, such as the acceleration due to gravity , is assumed to
be 10 ms² instead of 9.81 ms²
(e) ________________________________ in heatrelated experiments
4. Systematic errors can be ____________________ or _______________ by improving the
procedure of taking the measurements, using a ___________________________ or
______________________________________________.
5. Example of a good experimental procedure is to note the ____________________________,
such as micrometer screw gauge, stopwatches and ammeter before using the instrument. The
measurement is then corrected by ________________________ the zero reading from the
obtained readings.
Random Error
1. Random error refers to the scatter of reading about the ________________________.
2. The size of the random error is not ________________ and ___________________. It
deviates, sometimes ____________________ and sometimes __________________, from the
____________________________.
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3. Random error can be reduced by taking the mean value of _________________________ or
by _____________________________.
4. Some examples of random errors:
(a) ___________________________
(b) Change of conditions in the surroundings, such as __________________,
_______________ or _____________________.
(c) Nonuniform shape of an object such as ____________________________.
(d) Human reaction time.
Precision
1. The degree of a measuring instrument used to record consistent reading for each
measurement by the same way. It refers to the repeatability of a measurement. It does not
require us to know the correct or true value.
Accuracy
1. How close a measured value is to the actual value. In other words. Accuracy refers to the
degree of agreement between the experiment result and its true value.
Significant Figures
1. The number of digits known with uncertainty for a measured value is called the number of
_______________________________________.
2. The accepted convention is that only one digit uncertainty is to be reported for a
measurement. For example; if the estimated error is 0.01 m you would report a results as (0.53
± 0.01) m, not (0.528 ± 0.01) m
3. Students are frequently confused about when to count a zero as a significant figure. The rule
is if the zero has a nonzero digit anywhere on its left, then the zero significant, otherwise, it is
not. For example. 6.00 has 3 significant figure; the number 0.0006 has only
_______________________, and 2.0005 has ______________________________.
4. The number of significant figures for 400 is not well defined. Rather, one should write 4 x 102
___________________________, or 4.00 x 102, ________________________.
5. Normally, the number of significant figures that should be retained after
___________________ or __________________ of a number of quantities should follow the
significant number of the _____________ accurately known quantity.
6. For instance, the length and breadth (primary value) of an object 35.44 cm and 23.5 cm
respectively. The area of the object is calculated as,
35.44 cm x 23.5 cm
= 832.84 cm2
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=_____________ (follow the fewer number of significant figures of 23.5 cm)
7. For addition and subtraction of two or more values, the calculated answer should follow the
______________________________________ of the primary data. For instance,
28.366 cm + 1.5 cm
= 29.866 cm
=_____________ (follow the fewer number of decimal places of 1.5 cm)
Uncertainties in Measurement
1. Uncertainty of a measured value is an interval around that value such that any repetition of
the measurement will produce a new result that lies within this interval.
2. A length measured by a ruler normally can be recorded as 34.2 ± 0.1 cm. This means that the
uncertainty of the length measured by ruler is ± 0.1 cm and repetition of this measurement falls
between 34.1 cm and 34.3 cm.
3. Similarly a length measured by a vernier calipers normally can be recorded as 4.20 ± 0.01
cm.
4. A recorded value of 4.20 ± 0.01 cm gives us the confidence that the measured value lies
between 4.19 cm and 4.21 cm.
Determination of Uncertainties
1. Addition of Physical Quantities
(a) When there is addition of two physical quantities, we normally estimate the uncertainty
by adding the errors due to the two quantities.
(b) If X = P+ Q, then
ΔX = ΔP + ΔQ
(c) For instance, total current, I = I1 + I2
If I1 = (3.50 ± 0.02) A
and I2 = (6.50 ± 0.02) A
then I = [(3.50 ± 6.50) ± (0.02 + 0.02)]A
= (10.00 ± 0.04)A
2. Subtraction of Physical Quantities
(a) Similarly, when there is subtraction of two physical quantities, we can also estimate the
uncertainty for the difference of the two quantities by adding the uncertainties due to the
two quantities.
(b) If X = P – Q, then ΔX = ΔP + ΔQ
(c) For instance, temperature increase
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= final temperature – initial temperature
ϴ = ϴ1  ϴ2
If ϴ1 = (25.2 ± 0.1 ) ºC
ϴ2 = (65.4 ± 0.2) ºC
Then, = [(65.4 – 25.2)] ± (0.2 +0.1) ºC
= (40.2 ± 0.3 ) ºC
(d) Note that the addition of uncertainties is equal to (±0.3 ºC) instead of the subtraction of
uncertainties that is equal to ± 0.1ºC. This is to maximize the estimated uncertainty in a
calculated measurement.
3. Multiplication of Physical Quantities
(a) If X = AB², then
log X = log (AB²)
log X = log A + log B²
log X = log A + 2 log B
Differentiating the equation gives
∆ = ∆ + 2
In terms of uncertainties, we write
∆ = ± ∆ ±2
ΔX = (± ∆ ± 2 ) X
(b) For instance, area of a rectangle
= length x width, A = LW
If L = (50.0 ± 0.1 ) cm
W = (25.0 ± 0.1) cm
Then A = (50.0)(25.0) = 1250 cm²
∆ = ± ∆ ± ∆
∆ = ± 0.1 ± 0.1
1250 50.0 25.0
The uncertainty in area,
ΔA = ( ±500..10 ± 205.1.0) 1250 cm²
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= ±7.5 cm²
= ± 8 cm²
so, the recorded value of area = (1250 ± 8 ) cm²
4. Division of Physical Quantities
(a) If Y = , then
log Y = log C – log D
Differentiating the equation gives
∆ = ∆  ∆
To maximize the estimated uncertainty, we write
∆ =± ∆ ± ∆
ΔY = (±∆ ± )Y
(b) For instance, density = mass p = m
volume V
If m = (80.0 ± 0.2) g
v = (40.0 ± 0.2 ) cm³
then p = (40.0 ± 0.2) gcm³
then p = 80.0 = 2.00 gcm³
40.0
∆ = ± ∆ ± ∆
∆ = ± 0.2 ± 0.2
2.00 80.0 40.0
The uncertainty in density, Δp = ±0.015 = ±0.02 gcm³
the recorded value of density = (2.00 ± 0.02) gcm³
(c) Gives p = xy² ∆ = ∆ + 2∆ + ∆
z
The fractional error of p,
Percentage error of p,
∆ x 100% =( ∆ + 2∆ + ∆ )100%
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Eg:1
The following measurement were made to determine the density of a metal cylinder,
Diameter of cylinder, d = (2.46 ± 0.01) cm
Length of cylinder, l = (16.8 ± 0.1) cm
Mass of cylinder, m = (720 ± 10 )g
(a) What is the percentage error in the measurement of the density? [2.80%]
(b) Calculate the density of the metal to the correct number of significant figures.
(9.0 ± 0.3 )x 10³ kgm³
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