Questions & explanations
1. The length of a rod is measured as 125.67 cm. If the result should be reported with three significant figures, what is the correctly rounded value?
- 126 cm
- 125 cm
- 126.0 cm
- 125.7 cm
Answer: 126 cm
To express 125.67 cm with three significant figures, we round to the nearest whole number because the third digit is the units place. The digit after the third significant figure is 6 (≥5), so the third digit 5 increases to 6, giving 126 cm. Option B has three significant figures, but rounds 125.67 to the wrong precision (only 3 sig figs when we need to verify correct rounding — actually 125 is the truncated value, not properly rounded); it is incorrect because it drops the required rounding of the units digit, not because of sig fig count., while C and D both retain four significant figures.
2. The diameter of a wire measured with a screw gauge has a mean value of 2.34 mm and a mean absolute error of 0.02 mm. How should the result be expressed in standard form?
- (2.34 ± 0.02) mm
- (2.34 ± 0.002) mm
- 2.34 mm (error 0.02)
- (2.34 ± 0.2) mm
Answer: (2.34 ± 0.02) mm
The standard way to report a measurement with its uncertainty is to write (mean ± absolute error) with the appropriate units. Here, mean = 2.34 mm and absolute error = 0.02 mm, so the correct expression is (2.34 ± 0.02) mm. Option B uses an incorrect error magnitude (0.002), option C is not in standard parentheses format, and option D uses an error value (0.2) that does not match the given data.
3. If the equation v = a t + b/t is dimensionally correct, where v is velocity and t is time, then the dimensional formula of b is
- [M^0 L^1 T^0]
- [M^0 L^1 T^{-1}]
- [M^0 L^1 T^{-2}]
- [M^0 L^1 T^1]
Answer: [M^0 L^1 T^0]
According to the principle of homogeneity, each term in a dimensionally correct equation must have the same dimensions. The term b/t must have the dimensions of velocity, i.e., [L T^{-1}]. Since t has dimension [T], b must have dimension [L T^{-1}] × [T] = [L] = [M^0 L^1 T^0]. Option B corresponds to velocity, C to acceleration, and D to length×time, so only A is correct.
4. The dimensional formula for the gravitational constant G is
- [M^{-1} L^3 T^{-2}]
- [M L^3 T^{-2}]
- [M^{-1} L^2 T^{-2}]
- [M L^2 T^{-2}]
Answer: [M^{-1} L^3 T^{-2}]
From Newton's law of gravitation, F = G m1 m2 / r^2. Rearranging, G = F r^2 / (m1 m2). Force has dimensions [M L T^{-2}], r^2 has [L^2], and m1 m2 has [M^2]. Therefore G has dimensions ([M L T^{-2}] [L^2]) / [M^2] = [M^{-1} L^3 T^{-2}]. The other options have incorrect exponents for mass or length.
5. Which of the following physical quantities has the dimensional formula [M L^2 T^{-3}]?
- Force
- Energy
- Power
- Pressure
Answer: Power
Power is defined as energy per unit time. Energy has dimensions [M L^2 T^{-2}], so power has dimensions [M L^2 T^{-2}] / [T] = [M L^2 T^{-3}]. Force has [M L T^{-2}], energy has [M L^2 T^{-2}], and pressure has [M L^{-1} T^{-2}]. Hence only Power matches the given dimensional formula.
6. A vernier calliper has 20 divisions on the vernier scale that exactly coincide with 19 divisions of the main scale. Each main scale division is 1 mm. What is the least count of this instrument?
- 0.05 mm
- 0.10 mm
- 0.50 mm
- 0.01 mm
Answer: 0.05 mm
The least count is given by 1 main scale division (MSD) minus 1 vernier scale division (VSD). Since 20 VSD coincide with 19 MSD, 1 VSD = 19/20 MSD = 0.95 mm. Therefore, least count = 1 mm - 0.95 mm = 0.05 mm. This value indicates the measurement uncertainty of the instrument.
7. Which of the following cannot be determined using dimensional analysis?
- A) The dependence of time period on length and g
- B) The constant 2π in the pendulum formula
- C) The conversion factor between units
- D) The relation v = u + at
Answer: B) The constant 2π in the pendulum formula
Dimensional analysis can derive the form of a relationship up to a dimensionless constant, but it cannot provide the exact numerical value of dimensionless constants such as 2π. These must be obtained from experiments or theoretical considerations.
8. Which of the following equations is NOT dimensionally consistent?
- A) P = F v (power = force × velocity)
- B) P = F / a (power = force / acceleration)
- C) P = I^2 R (power = current^2 × resistance)
- D) P = V^2 / R (power = voltage^2 / resistance)
Answer: B) P = F / a (power = force / acceleration)
Power (P) has dimensions [M L^2 T^{-3}]. Force × velocity (Fv) has [M L T^{-2}][L T^{-1}] = [M L^2 T^{-3}], correct. I^2R and V^2/R also give power. But F/a: force/acceleration has dimensions [M L T^{-2}]/[L T^{-2}] = [M], which is not power.
9. A student measures the length of a table using a metre scale that has a zero error of +0.2 cm. The student also makes several measurements and obtains readings that vary slightly due to parallax. Which of the following correctly identifies the type of errors?
- Zero error is random, parallax error is systematic.
- Zero error is systematic, parallax error is random.
- Both zero error and parallax error are systematic.
- Both zero error and parallax error are random.
Answer: Zero error is systematic, parallax error is random.
Zero error is a constant offset that affects all measurements in the same way, making it systematic. Parallax error arises from the observer's position and tends to vary randomly, making it a random error.
10. The prefix 'nano' represents a factor of
- 10^{-6}
- 10^{-9}
- 10^{-12}
- 10^{-15}
Answer: 10^{-9}
In the SI system, the prefix 'nano' denotes a factor of 10^{-9} (one billionth). 10^{-6} corresponds to 'micro', 10^{-12} to 'pico', and 10^{-15} to 'femto'. Hence the correct factor is 10^{-9}.
11. Which of the following statements best distinguishes accuracy from precision?
- Accuracy refers to how close measurements are to each other, while precision refers to closeness to the true value.
- Accuracy is the degree of reproducibility, while precision is the closeness to the accepted value.
- Accuracy indicates the closeness of a measurement to the true value, while precision indicates the consistency of repeated measurements.
- Accuracy and precision are synonymous terms in measurement.
Answer: Accuracy indicates the closeness of a measurement to the true value, while precision indicates the consistency of repeated measurements.
Accuracy is the proximity of a measured value to the true or standard value, whereas precision is the repeatability or consistency of measurements when repeated under unchanged conditions.
12. The result of the addition 12.1 + 1.23 with appropriate significant figures is:
- 13.33
- 13.3
- 13
- 13.30
Answer: 13.3
In addition, the result should have the same number of decimal places as the least precise measurement. 12.1 has one decimal place, so the answer is rounded to one decimal: 13.3.