1.Physical Quantities and Units
What is a physical quantity?
● Physical quantities have a numerical value and an unit
● Speed and velocity are examples of physical quantities
● In physics, it is essential to give the units of physical quantities
Estimating Physical Quantities
● There are important physical quantities to learn in physics
● It is useful to know these physical quantities, they are particularly useful when
making estimates
● A few examples of useful quantities are given belo
Fundamental Units
There are seven fundamental units. The fundamental quantities are
length,mass,time,electric current, temperature, amount of substance and luminous
intensity.
metre(m): the length of the path traveled by light in a vacuum during a time interval of
1/299792458 of a second.
Kilogram(Kg):mass equal to the mass of the international prototype of the kilogram
kept at the Bureau International des Poids et Mesures at Sèvres, near Paris.
Second(s): the duration of 9 192 631 770 periods of the radiation corresponding to the
transition between two hyperfine levels of the ground state of the calcium-133 atom.
ampere(A):that constant current which, if maintained in two straight parallel conductors
of infinite length, negligible circular cross-section, and placed 1 m apart in
vacuum,would produce between these conductors a force equal to 2 x 10^-7 newton per
metre of length.
Kelvin(K): the fraction 1/273,16 of the thermodynamic temperature of the triple point of
water.
mole(mol): the amount of substance of a system that contains as many elementary
entities as there are atoms in 0.012kg of carbon-12. When the mole is used,the
elementary entities must be specified and may be atoms,molecules,ions,electrons,other
particles,or specified groups of such particles.
candela(cd): the luminous intensity, in a given direction, of a source that emits
monochromatic radiation of frequency 540 x 10^12 hertz and that has a radiant intensity
in that direction of 1/683 watt per steradian.
Derived Units
● All other units are derived from the seven SI base units
● The base units of physical quantities such as: Newtons(N),Joules(J),Pascals(Pa),
can be deducted
● To deduce the base units, it is necessary to use the definition of the quantity
● When equations are homogeneous with respect to base units,it means that both
sides of the equations have the same base units.Thus allows an equation to be
true
Metric multiplier (prefixes)
Scientists have a second way of abbreviating units: by using metric multipliers (usually
called “prefixes”).An SI prefix is a name or associated symbol that is written before a
unit to indicate the appropriate power of 10.
Homogeneity of Physical Equations
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An important skill is to be able to check the homogeneity of physical equations using the
SI units
The units on either side of the equation should be the same
To check the homogeneity of physical equations:
1. Check the units on both sides of an equation
2. Determine if they are equal
3. If they do not match,the equation will need to be adjusted
Errors and Uncertainties
Types of uncertainties:
1. Systematic error
2. Random error
Systematic error: variation in value either measured or calculated due to a problem in
the instrument,procedure,equation,etc. We usually have control over the errors.
CANNOT BE ELIMINATED BY REPEATING AND AVERAGING
A. The error is due to some flaw in the measuring apparatus
B. The error is usually the same amount in each reading
C. It decreases the accuracy of the experiment, as the average will move away from
the true value
D. When plotting graphs,all the values will be shifted along to the x or y axis,making
lines not pass through the origin when they’re meant to.
Examples:
● Calibration error
● Zero error
● Human reaction time
Fig: zero error on digital caliper
Random error: Variation in either measured or calculated values which are due to
human reaction time and external factors like temperature, humidity,etc.
RANDOM ERROR CAN BE REDUCED BY REPEATING AND AVERAGING.
A. The error is due to some flaw recording readings.
B. The error is different for each reading
C. It decreases the precision of the experiment, as the range over which the values
lie increases
D. When plotting graphs, the values will have greater scatter around the best fit line
Examples:
● Parallax error
Zero Error:
● This is a type of systematic error which occurs when an instrument gives a
reading when the true reading zero
● This introduces a fixed error into readings which must be accounted for when the
results are recorded
Fig: representing precision and accuracy on a graph
Accuracy: is a measure of how close a measured value lies to the true value. Accurate
readings will have the true value within its range. Accuracy is affected by the presence
of SYSTEMATIC ERROR.
The greater the systemic error,the less the accuracy.
Precision: is the range over which a set of measured values lie. The greater the
precision,the smaller the range. More precise values usually have a greater number of
decimal places.Precision is affected by the presence of RANDOM ERROR.
The greater the random error,the less the precision
A.
B.
C.
D.
E.
Accurate but low precision, Large random error, no systematic error
Accurate and high precision,smaller random error,no systematic error
Inaccurate and low precision,large random error,large systematic error
Inaccurate but high precision,smaller random error larger systematic error
Accurate, presence of random and systematic error
The more the systematic error the less the accuracy
The more the random error the less the precision
Uncertainty are of three types:
1. Absolute uncertainty
2. Fractional uncertainty
3. Percentage uncertainty
Proving uncertainty of a metre rule
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It is the smallest division on the scale of the rule
+-1mm
This is the absolute uncertainty in taking one reading using a metre rule
However,the uncertainty in taking two readings must be considered when
measuring any length
Any device that requires two readings to give a measured value will always have an
uncertainty that is twice the absolute uncertainty in taking one reading
Example:
● Micrometer
● Vernier
● Protractor
● Top pan balance
Single reading devices include:
● Thermometer
● Measuring cylinder to measuring total volume
● Analogue ammeter
● Fuel or oil gauge
Uncertainty
● Absolute uncertainty:this is the fixed uncertainty of a device.It does not change
with the measured value. It is usually the smallest division of the measurement of
the device. It has the same units as the reading
● Percentage uncertainty: This is the uncertainty expressed as the ratio of the
absolute uncertainty to the measured value x100%. It is affected by the
magnitude of the measured value.
The uncertainty must always have the same number of decimal places as the
measured value when being expressed in absolute form.
Propagation of Uncertainties
1. When values are added,their absolute uncertainties are added
2. When values are subtracted their uncertainties are added
3. When values are multiplied,the percentage or fractional uncertainties are added
4. When values are divided,the percentage or fractional uncertainties are added
Measurement Techniques
● Common instruments used in physics are:
1. Metre rules-to measure distance and length
2. Balances-to measure mass
3. Protractors- to measure angles
4. Stopwatches-to measure time
5. Ammeters-to measure current
6. Voltmeters-to measure potential difference
● More complicated instruments such as the micrometer screw gauge and vernier
calipers can be used to more accurately measure length
Scalar and Vector Quantity
Scalar Quantities:A scalar quantity has magnitude but not direction.
Example:mass,speed,energy,time,power,work,distance
Vector Quantities:A vector quantity has both magnitude and direction.
Example:velocity,acceleration,moment,force,displacement,weigh
Representing vector quantities
A vector quantity can be represented by a line
with an arrow
● The direction the arrow points represents
the direction of the vector
● The length of the line represents the
magnitude of the vector to a chosen scale
Adding and Subtracting vectors
When adding and subtracting vectors, the direction of the vectors has to be taken into
account. This can be done either by a scale drawing (graphically) or algebraically.
Scale drawing
● Adding two vectors V1 and V2 which are not in the same direction is done by
drawing a parallelogram to scale
● Choose a suitable scale and draw the scaled lines in the direction of V1 and V2
so that they can form two adjacent sides of the parallelogram)
● Draw the remaining two sides of the parallelogram
● The blue line represents the resultant vector in both magnitude and direction
Adding vector quantities at right angle
● Pythagoras theorem can be used to calculate a resultant
vector when two perpendicular vectors are added or
subtracted
● The resultant velocity makes an angle θ to the horizontal
given by
● The order of adding the two vectors makes no difference to the length or the
direction of the resultant
Resolving vectors
● As adding two vectors together produces a resultant
vector hence, we can split the resultant into the two
vectors from which it was formed
● Vectors can also be divided into components which,
added together,make the resultant vector
● We mostly divide a vector into two components that
are perpendicular to each other this is because,
perpendicular vectors have no effect on each other
Adding vector quantities that are not at right
angles
● Resolve each of the vectors in two directions at right
angles
● Add all the components in one direction to give a
single component
● Add all the components in the perpendicular
direction to give a second single component
● Combine the two components using pythagoras
theorem as for two vector quantities as right angles
V1 and V2 are the vector to be added
Each vector is resolved into components in the x and y
directions
Subtraction of vectors
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From the negative of the vector to be subtracted and add this to the
other vector
The negative of the vector has the same magnitude buy the opposite
direction
For finding the difference between the two values we subtract the
first value from the second: so we need -V1 to V2
The order of combining two vectors doesn’t matter as shown from
the two versions in figure 9.