Units and Measurements — Class 11 Physics
1. Physical Quantities
A physical quantity is a measurable property expressed by a numerical value and a unit. Examples include length, mass, time, velocity and force. A measurement is meaningful only when both the number and unit are stated.
2. SI System of Units
The International System of Units (SI) provides seven base quantities: length (metre, m), mass (kilogram, kg), time (second, s), electric current (ampere, A), thermodynamic temperature (kelvin, K), amount of substance (mole, mol) and luminous intensity (candela, cd).
Examples of derived units: velocity = m s−1, acceleration = m s−2, force = kg m s−2 = newton (N), and energy = kg m2 s−2 = joule (J).
3. Significant Figures
Significant figures indicate the meaningful precision of a measured quantity. Non-zero digits are significant. Zeros between non-zero digits are significant. Leading zeros are not significant. Trailing zeros after a decimal point are significant.
Example: 0.00450 has three significant figures: 4, 5 and the final 0.
4. Rules in Calculations
In multiplication or division, the result should generally contain the same number of significant figures as the quantity having the fewest significant figures. In addition or subtraction, the result is limited by the least precise decimal place.
5. Errors in Measurement
No physical measurement is perfectly exact. Absolute error describes the magnitude of deviation, while relative error compares the absolute error with the measured value. Percentage error is relative error multiplied by 100.
If measured values are x1, x2, … xn, the mean value is x̄=(x1+x2+…+xn)/n. Mean absolute error is the average of the absolute deviations from the mean.
6. Propagation of Errors
If z=x+y or x−y, the maximum absolute error in z is approximately Δz=Δx+Δy. If z=xy or x/y, the maximum fractional error is approximately Δz/z=Δx/x+Δy/y. If z=xn, then Δz/z≈|n|Δx/x.
7. Dimensions
Dimensions express a physical quantity in terms of base quantities. For example, velocity has dimensions [LT−1], acceleration [LT−2], force [MLT−2] and work [ML2T−2].
8. Dimensional Analysis
Dimensional analysis can check dimensional consistency, derive relations when the form of a relation is known, and convert units. It cannot determine dimensionless numerical constants such as 2 or π and cannot distinguish quantities having identical dimensions.
Worked Numerical 1 — Unit Conversion
Question: Convert 72 km h−1 into m s−1.
Solution: 72 km/h = 72 × (1000 m)/(3600 s) = 20 m s−1.
Check: The conversion factor 5/18 can also be used: 72×5/18=20.
Worked Numerical 2 — Significant Figures
Question: Calculate 2.5 × 3.42 with appropriate significant figures.
2.5×3.42=8.55. The first number has two significant figures, so the answer is 8.6.
Worked Numerical 3 — Percentage Error
Question: A length is measured as 5.00 m with an absolute uncertainty of 0.02 m. Find the percentage uncertainty.
Relative uncertainty = 0.02/5.00 = 0.004. Percentage uncertainty = 0.004×100 = 0.4%.
Worked Numerical 4 — Dimensions
Question: Find the dimensions of gravitational constant G from F=Gm1m2/r2.
G=Fr2/(m1m2). Therefore [G]=(MLT−2)L2/M2=[M−1L3T−2].
Conceptual MCQs — Explained
- Which of the following is an SI base unit?
A) newton B) joule C) kilogram D) pascal
Answer: C — kilogram. Newton and joule are derived units; kilogram is one of the seven SI base units. - How many significant figures are in 0.005060?
Answer: 4. The leading zeros are not significant; 5, 0, 6 and the final 0 are significant. - Which quantity has dimensions [ML2T−2]?
Answer: Energy/work. Work = force × displacement = [MLT−2]L. - Can dimensional analysis prove that a numerical equation is completely correct?
Answer: No. It can test dimensional consistency but cannot determine dimensionless constants or guarantee the physical validity of a relation. - If x has 2% error, what is the approximate percentage error in x2?
Answer: 4%. For z=xn, fractional error is approximately n times the fractional error in x.
Competency Question
A student records the diameter of a wire as 2.50 mm. Explain why writing “2.5 mm” communicates less information about measurement precision than “2.50 mm”. Then identify the number of significant figures in each.
HOTS
A student says that the equation v=u+at is correct because both sides have dimensions of velocity. Is dimensional consistency alone sufficient to prove the equation? Explain with reference to what dimensional analysis can and cannot establish.
Common Mistakes
- Writing a numerical answer without a unit.
- Treating every zero as significant.
- Forgetting that multiplying/dividing by a measured quantity affects uncertainty.
- Using dimensional analysis to determine numerical constants.
- Mixing kilometres and metres or hours and seconds in one calculation.
Quick Revision Sheet
- Velocity: [LT−1]
- Acceleration: [LT−2]
- Force: [MLT−2]
- Work/Energy: [ML2T−2]
- Power: [ML2T−3]
- Percentage error = (absolute error/mean measured value)×100
- For z=xn, Δz/z≈|n|Δx/x