1 Measurement

Question 1

What are physical quantities?

Answer 1

Consist of a numerical magnitude and a unit

Question 2

What are base quantities? How many of them are there? What are they?

Answer 2

The 7 base quantities are physical quantities that are the most fundamental and they are used to define other physical quantities. They consist of:

Length - meter (m)
Time - second (s)
Amount of substance - mole (mole)
Electric current - ampere (A)
Temperature - kelvin (K)
Luminous intensity - candela (cd)
Mass - kilogram (kg)

Question 3

What are derived quantities? How are they expressed?

Answer 3

Derived quantities are physical quantities formed by combining base quantities according to a defining equation that relates the physical quantities/algebraic relations involving products and/or quotients. They are defined in terms of base units and are expressed as products and/or quotients of base units.

Question 4

What are some examples of derived quantities?

Answer 4

Force = mass times acceleration
Units of Force = kgms^-2 = kg m s^-2
Work done = force times displacement
Units of Work done = kg m s^-2 *m = kg m^2 s^-2

Question 5

What does the homogeneity of equations mean?

Answer 5

In order for a physical equation to be operational or meaningful, each term in the equation must have the same base units. This is because only quantities with the same base units can be added, subtracted or equated. When this is fulfilled, the equation is said to be homogeneous or dimensionally consistent.

Question 6

Are all homogeneous equations physically correct? When might a homogeneous equation be physically wrong?

Answer 6

Homogeneous equations need not be physically correct as they may have missing terms or wrong coefficients or signs.

Question 7

What is a dimensionless constant?

Answer 7

A constant with no unit.

Question 8

Is it possible to deduce the value of constant through the homogeneity of equations?

Answer 8

No, constants can only be determined through more rigorous derivations or experimentation.

Question 9

How can we indicate decimal sub-multiples or multiples of both base and derived units?

Answer 9

Using prefixes and their symbols:
(refer to pg 6 of notes)
kilo, mega, giga, tera
deci, centi, milli, micro, nano, pico

Question 10

What is an order of magnitude?

Answer 10

We refer to an order of magnitude estimate of a physical quantity as the power of ten of the number that describes it. If two numbers differ by one order of magnitude, one is about 10 times larger than the other. If two numbers have the same order of magnitude, the larger value is less than ten times the smaller value. (State that the order of magnitude is 10^x at A levels instead of just x)

Question 11

What are the 7 rules to determine the number of significant figures?

Answer 11

All non-zero numbers are significant.
Zeros between two non-zero digits are significant.
Leading zeros are not significant.
Trailing zeros to the right of the decimal are significant.
Trailing zeros in a whole number with the decimal shown are significant.
Trailing zeros in a whole number with no decimal shown are not significant.
Exact numbers have an infinite number of significant figures.

Question 12

Regarding measuring instruments and their inherent uncertainty, all measuring instruments and methods of measurement are said to have limitations hence it is important to express their measurements together with their uncertainties. How do we determine the uncertainty of a measuring instrument?

Answer 12

The inherent uncertainty when reading off the scale of a measuring instrument is determined by the precision of the instrument which is also the smallest division of the scale.

Question 13

What does it mean for a measuring instrument to be more precise than another? What is the precision of the metre rule, vernier caliper and micrometer screw gauge?

Answer 13

As the precision of a measuring instrument increases, the number of significant figure quoted by the instrument increases and is more precise than another reading quoted by a less precise instrument.

Metre rule - 0.1cm or 1mm
Vernier caliper - 0.01cm or 0.1mm
Micrometer screw gauge - 0.001cm or 0.01mm

Question 14

What are systematic errors? What are some examples of systematic errors? How can they be eliminated? Can it be eliminated by repeating measurements and averaging them?

Answer 14

Systematic errors result in all readings or measurements being either always above or always below the TRUE value by a fixed amount. Some examples include not accounting for zero error in measurement or background radiation. It can only be eliminated only if the source of the error is known and accounted for. It cannot be eliminated by repeating measurements and averaging them.

Question 15

What are random errors? What are some examples of random errors? How can they be eliminated?

Answer 15

Random erros result in all readings or measurements being scattered about a mean value. These errors have equal probability of being positive or negative. Some examples include variation in diameter of a piece of wire when measurements are taking along the piece of wire or fluctuations in the count-rate if radioactive decay. It can be eliminated by repeating measurements and finding the average value or by plotting a graph and drawing a line of best fit for the plotted points.

Question 16

Define accuracy. Is it affected by systematic or random errors? What does it mean when a set of data is highly accurate?

Answer 16

Accuracy is the degree of closeness of the MEAN value of the measurements to the true value. It is affected by systematic errors. A highly accurate data set means that the mean value centers around the true value.

Question 17

Define precision. Is it affected by systematic or random errors? What does it mean when a set of data is highly precise?

Answer 17

Precision is the degree of agreement between repeated measurements of the same quantity. It is affected by random errors. A highly precise data set means that there is a small spread of data.

Question 18

What 4 factors of a measuring instrument affect its accuracy and precision?

Answer 18

Responsiveness of an instrument is the ability of the instrument to detect rapid variations in the quantity being measured.

Sensitivity of an instrument is the ability of the instrument to detect small changes in the quantity being measured.

Reliability is the measure of confidence that can be placed in a set of measurements (accurate AND precise)

Reproducibility

Question 19

How can we calculate uncertainties of derived quantities? How must uncertainty always be represented?

Answer 19

Method 1: Upper-lower bound method of ucnertainty propagation (useful when the relationship between variables is complex or incomplete.)

It uses the uncertainty ranges of each measurement to calculate thhe maximum and minimum values of the derived quantity and then using them to determine the uncertainty of the derived quantity where Uncertainty = (Qmax - Qmin) / 2

Method 2: Uncertainty formulae (Refer to pg 13 of notes)

(i) Addition and Subtraction (Absolute uncertainty)
(ii) Product and Division (Fractional uncertainty)
(iii) Product or Quotient with powers (Fractional uncertainty)

Uncertainty must always have 1 significant figure and the reading and uncertainty must have the same number of decimal places.

They do not cancel each other out, they always add up to maximise the uncertainty.

Always rearrange the equation so that the variable of interests apears on the left-hand side.

Question 20

How is the percentage uncertainty calculated?

Answer 20

Fractional uncertainty times 100%

Question 21

What is the difference between scalar and vectors?

Answer 21

Scalar quantity has magnitude only.

Vector quantity has both magnitude and direction.

Question 22

How is the resultant vector calculated?

Answer 22

Vector addition

Method 1: Tip to tail AKA Polygon law
Joining the head of one vector to the tail of another Magnitude can be found using the cosine rule
Direction is found using the sine rule)

Method 2: Parallelogram law

Vector substraction
Method: Rewriting vectors
(Flipping in the opposite direction)

Question 23

What are the steps to carry out when resolving vectors?

Answer 23

1. Choose orientation of x-y coordinate system
2. Label angles
3. Drop perpendicular to axes
4. Apply trigo to find components (hyp must always be the original vector and components must always be shorter)

For multiple vectors,

5. Determine the resultant components along the x- and y-axis
6. Apply Pythagoras theorem to find magnitude of the resultant vector
7. Apply trigo to find the angle that the resultant vector makes with the x- or y- axis or another other reference line