Units and Measurements: common doubts

Question 1

What is the difference between a base unit and a derived unit?

Accepted Answer

A base unit is chosen and defined on its own, while a derived unit is built by combining base units. SI picks seven base quantities: length, mass, time, electric current, temperature, amount of substance and luminous intensity. Everything else, such as the newton (kg m s⁻²) or the joule (kg m² s⁻²), follows from these through a defining equation, so only seven standards ever need to be maintained.

Question 2

What are the seven SI base units?

Accepted Answer

They are the metre (length), kilogram (mass), second (time), ampere (electric current), kelvin (thermodynamic temperature), mole (amount of substance) and candela (luminous intensity). Every other SI unit can be written as a product of powers of these seven. The radian and steradian are dimensionless units for plane and solid angle and are not counted among the base units.

Question 3

Why is the radian a unit but dimensionless?

Accepted Answer

The radian is defined as arc length divided by radius, dθ = ds/r, so it is a ratio of two lengths and its dimensions cancel. It is still named so that we know the number refers to an angle measured in radians rather than degrees. The steradian, dΩ = dA/r², is dimensionless in the same way because it divides an area by a squared length.

Question 4

How many significant figures does 0.00230 have?

Accepted Answer

It has three significant figures: 2, 3 and the final 0. The zeros before the 2 only place the decimal point and would vanish if the value were written as 2.30 × 10⁻³. The trailing zero after the decimal point is significant because someone deliberately recorded it, showing the measurement was precise to that digit.

Question 5

Are trailing zeros in a number like 4700 significant?

Accepted Answer

In a bare number like 4700 they are ambiguous, which is why scientific notation is used to remove the doubt. Writing 4.7 × 10³ shows two significant figures, while 4.700 × 10³ shows four. A change of unit cannot add precision, so 4.7 m converted to 4700 mm still has only two significant figures and should be written 4.7 × 10³ mm.

Question 6

Why does scientific notation make significant figures clear?

Accepted Answer

Because in the form a × 10ᵇ with 1 ≤ a < 10, every digit shown in a is significant and the power of ten only sets the size. The zeros that merely locate the decimal point move into the exponent. So 0.0052 becomes 5.2 × 10⁻³ with two significant figures, and a choice of unit no longer changes the count.

Question 7

What is the rule for significant figures when adding or subtracting?

Accepted Answer

The result keeps as many decimal places as the measurement with the fewest decimal places. Addition is limited by the least precise absolute position, not by the count of significant figures. For example, 436.32 + 227.2 + 0.301 gives 663.821, which is reported as 663.8 because 227.2 is known only to one decimal place.

Question 8

What is the rule for significant figures when multiplying or dividing?

Accepted Answer

The answer keeps the same number of significant figures as the factor with the fewest significant figures. A product cannot be known more precisely, in relative terms, than its roughest input. So 4.237 g divided by 2.51 cm³ is reported to three significant figures, 1.69 g cm⁻³, even though a calculator shows many more digits.

Question 9

Do exact numbers like 2 in 2πr limit significant figures?

Accepted Answer

No, exact numbers have unlimited significant figures and never restrict the answer. The 2 in 2πr, or a count such as 20 oscillations used to find one period, is not a measurement and carries no uncertainty. Only measured quantities decide the precision of the result, so ignore pure counting numbers and defined factors when applying the significant-figure rules.

Question 10

How do you round off a number ending in 5?

Accepted Answer

If the digit to be dropped is exactly 5 with nothing after it, leave the preceding digit unchanged when it is even and raise it by one when it is odd. So 2.745 becomes 2.74, while 2.735 becomes 2.74 as well. This even-digit convention avoids always pushing results upward, which would bias a long set of rounded values.

Question 11

How do errors combine when two measured quantities are multiplied?

Accepted Answer

The relative errors add: if Z = A × B or Z = A/B, then ΔZ/Z = ΔA/A + ΔB/B. Absolute errors are not added for products and quotients, because each factor scales the other's uncertainty. For a power, Z = Aⁿ, the relative error becomes n times ΔA/A, which is why a quantity raised to a high power needs the most careful measurement.

Question 12

Why does the same instrument give a larger percentage error for a smaller reading?

Accepted Answer

Because the uncertainty of a reading is fixed by the instrument's least count, while the percentage error divides that uncertainty by the value measured. A balance good to ±0.01 g gives about ±1% on a 1.02 g sample but only about ±0.1% on 9.89 g. So small quantities need a finer instrument, or a larger quantity can be measured and divided, as when timing 20 oscillations instead of one.

Question 13

What is the difference between dimensions and units?

Accepted Answer

Dimensions tell you which base quantities a physical quantity is built from and with what powers, while units are the particular standard used to express its size. Force has dimensions [M L T⁻²] whether it is measured in newtons or dynes. Units change with the system you pick; dimensions do not, which is what makes dimensional checks useful.

Question 14

What is the dimensional formula of density?

Accepted Answer

Density is mass divided by volume, so its dimensional formula is [M L⁻³ T⁰]. Mass contributes M to the first power and volume contributes L³ in the denominator, while time does not appear at all. Writing T⁰ explicitly is optional but shows clearly that time plays no part in the quantity.

Question 15

If both sides of an equation have the same dimensions, is it correct?

Accepted Answer

Not necessarily; dimensional consistency is required but does not prove an equation right. A dimensional check cannot detect a wrong numerical factor, so x = at² passes just as well as x = ½at². It also cannot catch a wrong term that happens to have the right dimensions. An equation that fails the check, however, is certainly wrong.

Question 16

Why can you only add quantities that have the same dimensions?

Accepted Answer

Because adding a length to a time, or a force to an energy, has no physical meaning. Every term in a valid equation must therefore carry identical dimensions, which is called the principle of homogeneity. This gives a quick test: in v² = u² + 2as each term has dimensions [L² T⁻²], so the equation passes the check.

Question 17

What are the limitations of dimensional analysis?

Accepted Answer

It cannot find dimensionless constants such as the 2π in T = 2π√(l/g), and it cannot handle relations involving trigonometric, exponential or logarithmic functions. It cannot derive a relation that is a sum of terms, such as s = ut + ½at², and in mechanics it fails when a quantity depends on more than three variables, since only M, L and T are available to match.

Question 18

How do you derive the period of a pendulum using dimensions?

Accepted Answer

Assume T is proportional to lᵃ gᵇ mᶜ and match the powers of M, L and T on both sides. Mass appears nowhere else, so c = 0. Matching L gives a + b = 0 and matching T gives −2b = 1, so b = −½ and a = ½. This yields T ∝ √(l/g); the constant 2π must come from experiment or a full derivation.

Question 19

Can two different physical quantities have the same dimensions?

Accepted Answer

Yes. Work and torque both have dimensions [M L² T⁻²], yet torque is a turning effect, not an energy. Pressure and stress share [M L⁻¹ T⁻²], and frequency and angular velocity share [T⁻¹]. So matching dimensions only show that two quantities could be compared or added; they do not prove the quantities are the same thing. Dimensional analysis alone cannot tell such pairs apart.

Question 20

Can a quantity have units but no dimensions?

Accepted Answer

Yes. Plane angle and solid angle are the standard examples: they are measured in radians and steradians, yet each is a ratio of like quantities and so is dimensionless. Strain is dimensionless and is usually quoted without a unit. The reverse cannot happen, since anything with dimensions needs a unit to express its size.

Units and Measurements: common doubts, answered

About the chapter

Can a quantity have units but no dimensions?

Units: base, derived and systems