Conceptual Questions

A good physical standard must be: Accessible (reproducible worldwide), Invariable (unchanging over time and location), Precise (allows measurement to desired accuracy), and Reproducible. These are essential because science relies on verifying results. If standards varied, experiments in different labs or at different times would not be comparable, hindering scientific progress.

The transition provides several advantages: Universal accessibility (no need to compare with a physical object in Paris), Ultimate stability (fundamental constants don't change or degrade like physical artifacts), and Improved precision for future technologies. It makes the SI system truly universal and future-proof.

Dimensional analysis works by ensuring dimensional consistency, which constrains the powers of variables but not dimensionless constants. For example, for a pendulum, dimensional analysis correctly finds that the period T ∝ √(l/g). However, it cannot determine the constant of proportionality, which is 2π. This constant must be found through experiment or more advanced mathematical derivation.

Yes. Dimensional consistency is necessary but not sufficient. Example: For kinetic energy, both K = mv² and K = ½mv² are dimensionally correct ([ML²T⁻²]). However, only the second formula is physically correct. Dimensional analysis can't find the error in the first formula because the missing factor (1/2) is dimensionless.

You would check the dimensions. The dimension of force F is [MLT⁻²]. The dimension of viscosity η is [ML⁻¹T⁻¹].
Proposed formula: [r²v] = [L²][LT⁻¹] = [L³T⁻¹]. This is incorrect.
Correct formula: [ηrv] = [ML⁻¹T⁻¹][L][LT⁻¹] = [MLT⁻²]. This matches the dimensions of force. The proposed formula is dimensionally inconsistent.

Accuracy refers to how close measurements are to the true value. Precision refers to how close repeated measurements are to each other.
Systematic errors (e.g., a miscalibrated scale) affect accuracy, causing all measurements to be consistently off-target.
Random errors (e.g., fluctuations in reading a scale) affect precision, causing measurements to be scattered.

These functions are defined by their infinite series expansions (e.g., eˣ = 1 + x + x²/2! + ...). For the sum to be valid, every term must have the same dimension. If x had a dimension (e.g., length [L]), then the terms would have dimensions of [1], [L], [L²], etc., which cannot be added. Therefore, the argument x must be dimensionless.

The measurement of the radius r is the sole contributor to the error. The value of π is a defined mathematical constant, not a measured quantity, and is considered to have infinite precision (it's an exact number). The error propagation formula ΔA/A = 2(Δr/r) shows that the relative error in area is twice the relative error in the radius measurement.

This distinction helps identify the source of measurement problems. High precision with low accuracy suggests a systematic error that needs to be found and corrected. Low precision suggests significant random error, which might be improved by using a better instrument or refining the measurement technique (e.g., taking more data points and averaging).

In addition/subtraction, the uncertainty is determined by the absolute position of the last significant digit (the decimal place). The result cannot be more precise than the least precise number (e.g., adding a value known to the thousandths place to one only known to the tenths place is limited by the tenths place). In multiplication/division, the uncertainty is determined by the relative error, which is related to the number of significant figures. The result's relative uncertainty is limited by the number with the largest relative uncertainty (i.e., the fewest significant figures).

It is ambiguous because we don't know if the trailing zeros are placeholders or are actually measured significant digits. It could mean 2, 3, or 4 significant figures. The ambiguity is removed by using scientific notation:
• 1.2 × 10³ m (2 significant figures)
• 1.20 × 10³ m (3 significant figures)
• 1.200 × 10³ m (4 significant figures)

You add absolute errors when the quantities are being added or subtracted. For example, if Z = A + B, then ΔZ = ΔA + ΔB.
You add relative errors when the quantities are being multiplied or divided. For example, if Z = A × B, then ΔZ/Z = ΔA/A + ΔB/B.

Relative error is generally a better indicator. An absolute error of 1 cm is very large when measuring the width of a book (e.g., 20 cm), but it is exceptionally small when measuring the distance to the moon. Relative error puts the error into context by comparing it to the magnitude of the measurement itself, providing a better assessment of its quality.

This is due to the power rule of error propagation: ΔZ/Z = 3(ΔA/A). The percentage error in Z is three times the percentage error in A. The cubic relationship amplifies the effect of the initial measurement error. For example, a 2% error in measuring the radius of a sphere leads to a ~6% error in calculating its volume.

Yes, it is possible. It signifies an extremely poor measurement where the uncertainty (ΔA) is larger than the measured value itself (A). For example, a result of (2 ± 3) cm has a relative error of 1.5. This means the "true" value is so uncertain it could even be negative (which may be unphysical), and the measurement provides very little useful information.

No. While they are fundamental constants of nature, they are not fundamental quantities in the dimensional sense. Their dimensions are derived from the seven base SI quantities (Mass, Length, Time, etc.).
• Dimension of h is [ML²T⁻¹].
• Dimension of c is [LT⁻¹]. The fundamental quantities are the base dimensions themselves ([M], [L], [T], etc.).

The laser interferometer measurement will have more significant figures and a smaller percentage error. The precision of the measuring instrument determines the number of significant figures you can confidently report. The laser, being a much more precise instrument, can resolve smaller length increments, leading to more significant figures and a smaller uncertainty (ΔL), which in turn leads to a smaller percentage error (ΔL/L).

Dimensional analysis fails for formulas involving sums or differences. The method works by assuming a product relationship like v = k × uᵃ × aᵇ × tᶜ. While it can confirm that all terms in v = u + at have the same dimensions ([LT⁻¹]), it cannot generate a formula that is a sum of terms. This is one of its key limitations.

No, this claim is incorrect. While systematic errors can theoretically be identified and eliminated, random errors are inherent in any measurement process. There will always be limitations in the measuring instrument (its least count) and unavoidable random fluctuations in the environment or the observer's perception. Therefore, every measurement has some degree of uncertainty that can be minimized but never completely eliminated.

Precision relates to the size of the uncertainty. Student B's uncertainty is ±1 Ω, while Student A's is ±10 Ω. Therefore, Student B was more precise.
Accuracy relates to how close the result is to the true value. Student B's range (104-106 Ω) includes the true value. Student A's range (90-110 Ω) also includes the true value, but Student B's central value (105 Ω) is much closer to the true value (106 Ω) than Student A's (100 Ω). Therefore, Student B was also more accurate.