Measurements
Multiple Choice Questions with Explanations
1.1 The purpose of study and discoveries in Physics is:
(a) the probing of interstellar spaces
(b) the betterment of mankind ✓
(c) the development of destructive technology in warfare
(d) development in aesthetics for the world
✅ Explanation: Physics ka asal maqsad insani zindagi ko behter banana hai — technology, health, communication, transport etc. sab is ka result hain.
1.2 The length of a steel pipe is in between 0.7 m to 0.8 m. Identify from the
following, the appropriate instrument to be used for an accuracy of 0.001 m.
(a) A micrometer screw gauge
(b) A metre rule
(c) A ten metres measuring tape
(d) A Vernier Callipers ✓
✅ Explanation: Vernier Callipers ki accuracy 0.01 cm (yaani 0.0001 m) tak hoti hai, is liye steel pipe ki measurement mein yeh best instrument hai.
1.3 The diameter of a steel ball is measured using a Vernier callipers and its
reading is shown in the figure. What is the diameter of the steel ball?
(a) 1.30 cm
(b) 1.39 cm ✓
(c) 1.40 cm
(d) 1.31 cm
✅ Explanation: Agar main scale 1.3 cm aur vernier scale ka 9th division match kar raha ho to:
1.3 + (9 × 0.01) = 1.39 cm
1.4 A micrometer screw gauge is used to measure the diameter of a rod. One complete turn of the thimble = 0.50 mm and 50 divisions on the circular scale.
(a) 3.67 mm
(b) 3.17 mm ✓
(c) 4.77 mm
(d) 4.20 mm
✅ Explanation:
Pitch = 0.50 mm
LC (Least Count) = 0.50 mm / 50 = 0.01 mm
Assume main scale reading = 3.0 mm, and circular scale = 17
→ Diameter = 3.0 + (17 × 0.01) = 3.17 mm
1.5 The number of significant figures of a measurement are defined as:
(a) they reflect the accuracy of the observation in a measurement
(b) they are the figures which are reasonably rounded off
(c) they are the accurately known digits and the first doubtful digit of a measurement ✓
(d) meaningless data
✅ Explanation:
Significant figures include all known digits plus the first uncertain (doubtful) digit.
1.6 The number of significant figures in the measured mass 2500.0 kg is:
(a) two
(b) three
(c) four
(d) five ✓
✅ Explanation:
All non-zero digits and trailing zeroes after decimal are significant. So:
2500.0 → 5 significant figures
1.7 The sum 12 kg + 2.02 kg + 5.1 kg, to appropriate precision:
(a) 19 kg
(b) 19.0 kg
(c) 19.1 kg ✓
(d) 19.12 kg
✅ Explanation:
Least precise number (12 kg) has no decimal, but 5.1 has 1, and 2.02 has 2 decimal places. Final answer should match the least decimal places (i.e., 1).
12 + 2.02 + 5.1 = 19.12 → round to 19.1
1.8 Subtraction: 0.126 – 0.97268 = ? (to correct significant figures)
(a) 0.15
(b) 0.153
(c) 0.1533 ✓
(d) 0.15332
✅ Explanation:
0.126 has 3 decimal places, so answer should be up to 3 decimal places → 0.1533
1.9 Product: 2.8723 × 1.6 = ? (Significant figures)
(a) 4.59568
(b) 4.595
(c) 4.59 ✓
(d) 4.6
✅ Explanation:
2.8723 → 5 sig. figs
1.6 → 2 sig. figs
So answer must be in 2 significant figures → 4.59
1.10 Division: 45.2 ÷ 6.0 = ?
(a) 7.5 ✓
(b) 7.53
(c) 7.533
(d) 7.5333
✅ Explanation:
45.2 (3 sig. figs), 6.0 (2 sig. figs) → final result = 2 sig. figs → 7.5
1.11 Multiplication: 24.4 m/s × 100 m × 2.0 s
(a) 4880 m
(b) 4900 m ✓
(c) 4.88 × 10³ m
(d) 4.9 × 10³ m
✅ Explanation:
Significant figures:
24.4 (3), 100 (1), 2.0 (2) → answer must have 1 sig. fig (least)
→ 4880 → rounded to 4900 (1 sig. fig)
1.12 The ratio of the dimensions of force and energy is:
(a) T
(b) T⁻¹
(c) L
(d) L⁻¹ ✓
✅ Explanation:
Force=MLT−2, Energy=ML2T−2
So ratio = Force / Energy = MLT−2 / ML2T−2 =1 / L= L−1
1.13 Which pair does not have identical dimensions?
(a) Work and torque
(b) Angular momentum and Planck’s constant
(c) Moment of inertia and moment of force ✓
(d) Impulse and momentum
✅ Explanation:
Moment of inertia: ML²
Moment of force (Torque): M L² T⁻² → different dimensions
1.14 The following figures are of the same Vernier Calipers.
Fig (1) shows the reading when the jaws are closed
Fig (2) shows the reading when a solid cylinder is placed between the jaws
the length of the cylinder?
Options:
(a) 3.26 cm
(b) 3.30 cm
(c) 3.34 cm ✓
(d) 4.20 cm
1.15 The least count of an instrument determines:
(a) Precision of a measurement ✓
(b) Accuracy of a measurement
(c) Fractional uncertainty
(d) Percentage uncertainty
✅ Explanation:
Least Count shows the smallest measurable quantity. It reflects how precise the instrument is.
1.16 A measuring tape has a smallest division of 0.02 cm. The appropriate reading using this tape is:
(a) 80.504 cm
(b) 80.6 cm
(c) 80.65 cm ✓
(d) 80.7 cm
✅ Explanation:
A measuring tape with 0.02 cm smallest division can report values up to 2 decimal places, so
80.65 cm is valid.
� Short Answer Questions
Q1.1 What are base units and derived units? Give examples.
Answer:
Base units are the fundamental units for physical quantities (e.g., metre, kilogram,
second).
Derived units are obtained by combining base units (e.g., Newton = kg·m/s², Joule =
kg·m²/s²)
Q1.2 How many significant figures should be retained?
(i) Multiplication/Division: Keep the least number of significant figures among all
values.
(ii) Addition/Subtraction: Keep the result with the least decimal places.
Q1.3 How is the Vernier scale related to the main scale? What is L.C.?
Answer:
1. Vernier scale is a small sliding scale used for precise measurements.
2. L.C. (Least Count) is the smallest value that can be measured using the Vernier.
LC=Value of one main scale division − Value of one vernier scale division
Q1.4 Write in scientific notation:
(a) 143.7 → 1.437×10^2
(b) 2126.4 × 10⁷ → 2.1264×10^10
Q1.5 Use correct prefixes:
(a) 580 × 10³ g = 580 kg
(b) 0.454 × 10⁴ s = 4.54 × 10³ s = 4.54 ks
Q1.6 What are the dimensions of Kinetic Energy (½mv²)?
Answer:
[K.E.]=1 / 2 [M][L2][T^−2]=[ML2T^−2]
Q1.7 How many significant figures?
(i) 37 km → 2
(ii) 0.032953 m → 5
(iii) 7.50034 cm → 6
(iv) 200.0 m → 4
Q1.8 Dimensions of:
(i) Planck’s constant h → [M L² T⁻¹]
(ii) Angular velocity → [T⁻¹]
Constructed Response Questions
Q1.1 Why do we use both kilogram and mole for amount of substance?
Answer:
Kilogram is used to measure mass.
Mole is used to count entities (atoms/molecules).
Both describe amount but in different contexts.
Q1.2 Three students measured a rod using scale (LC = 1 mm):
(i) 0.423501 m ❌
(ii) 0.42 m ✓
(iii) 0.424 m ✓
Conclusion:
Only (ii) and (iii) are correct measurements according to the scale’s precision.
(i) is invalid because it records more digits than the instrument can measure.
Explanation: First value is over-precise (more than instrument’s capacity).
Q1.3 Why is kilogram (not gram) the base unit of mass?
Answer:
The kilogram (kg) is the base unit of mass in the International System of Units (SI) instead of
the gram because:
- ✅ Historical Reason:
The SI system was established based on the metric system, in which the kilogram was
originally defined using a physical prototype (the International Prototype Kilogram or
IPK) in 1889. - ✅ Practical Usage:
The kilogram is more suitable for measuring everyday quantities of mass (like human
weight, commercial goods, scientific experiments).
Using gram as the base unit would result in very large numbers for most real-world
masses. - ✅ Scientific Standardization:
In 2019, the kilogram was redefined based on fundamental constants (Planck’s
constant), not a physical object — reinforcing it as the most stable and precise unit for
mass.
�
� Conclusion:
Although gram is a smaller and widely used unit, the kilogram is adopted as the base unit due
to historical, practical, and scientific reasons.
Q1.4 If P = Q × R, and both Q & R have dimensions of velocity ([LT⁻¹]),
Answer:
�
� Step 1: Find dimensions of P
Given:
[Q]=[LT−1], [R]=[LT−1]
[P]=[Q]⋅[R]=[LT−1]⋅[LT−1]=[L^2T^−2 ]
�
� Step 2: Find SI units of P
From dimensions:
[L2T−2]=Unit of Energy or Work
✅ SI Unit of P = Joule (J)
Where:
1 Joule=1 kg\cdotpm2/s^2
�
� Step 3: If Q and R had different dimensions
Yes, even if Q and R had different dimensions, we can still determine the dimensions of P by
multiplying their dimensions algebraically.
�
� Example:
If
[Q]=[M], [R]=[LT^−2]⇒[P]=[MLT^−2
Which are the dimensions of force (Unit: Newton)
1.5
Q: What is the least count of a clock if it has…
(a) Hour, minute, and seconds hand?
�
� Least Count = 1 second
Because the seconds hand moves every 1 second — that’s the smallest measurable time.
(b) Hour and minute hand only?
�
� Least Count = 1 minute
Since there is no seconds hand, the smallest observable time is when the minute hand moves,
i.e., 1 minute.
1.6
Q: How can diameter of a round pencil be measured using a meter rule with same accuracy as Vernier Calipers?
Answer:
Place the pencil between two blocks (like fingers) and mark the edges. Measure distance
between marks using a meter rule 3–4 times and take average. Accuracy increases by repetition.
1.7
Q: How would readings differ if screw gauge is used instead of Vernier Calipers to measure thickness of glass plate?
Answer:
Screw gauge has higher accuracy (LC = 0.01 mm) than Vernier Calipers (LC = 0.1 mm). So,
readings would be more precise with screw gauge.
Reading with error:
If measured value is 3.34 cm and error is ±0.02 cm,
→ Corrected value = 3.34 ± 0.02 cm
1.8
Q: Screw gauge has 50 divisions. If thimble is rotated 10 times, and spindle moves 5 mm. What is pitch and LC?
Answer:
Pitch = 5 mm / 10 = 0.5 mm
LC = 0.5 mm / 50 = 0.01 mm
1.9
Q: Screw gauge shows zero error (positive), and 2nd circular division matches datum line.
Answer:
Zero error = +0.02 mm
Suppose observed reading = 4.57 mm
→ Correct reading = 4.57 mm – 0.02 mm = 4.55 mm
1.10
Q: What is meant by a dimensionless quantity? Give one example.
Answer:
A physical quantity having no units or dimensions is called dimensionless.
� Example: Angle (radian), refractive index
1.11
Q: A sheet of paper folded 3 times is measured by screw gauge. Total thickness = 0.36 mm
Answer:
No. of sheets = 2³ = 8 layers
→ Thickness of one sheet = 0.36 mm / 8 = 0.045 mm
1.12
Round off to 3 significant figures & convert to scientific notation:
(a) 0.02055 → 2.06 × 10⁻²
(b) 4656.5 → 4.66 × 10³
Comprehensive Questions
Comprehensive Questions
Q1.1
What is meant by uncertainty in a measurement? How is uncertainty in digital instruments indicated?
Answer:
Uncertainty in Measurement:
Uncertainty in a measurement refers to the doubt or margin of error associated with the result. It
represents the range of values within which the true value is likely to lie.
Types of Uncertainty:
- Random Uncertainty: Due to unpredictable fluctuations in measurements.
- Systematic Uncertainty: Due to instrumental or methodological errors.
Uncertainty in Digital Instruments:
Digital instruments typically display a finite number of digits. The uncertainty is often indicated
by: - Last Digit: The last digit displayed is usually considered uncertain.
- Resolution: The smallest change in measurement that can be detected.
For example, a digital multimeter displaying 4.52 V might have an uncertainty of ±0.01 V.
Reporting Uncertainty:
Measurements are often reported with uncertainty, e.g., 4.52 V ± 0.01 V.
Q1.2
Differentiate between precision and accuracy: Feature Precision Accuracy
Definition Reproducibility of measurements Closeness to true value
Example Same wrong value repeatedly One correct value, even once
Q1.3
(a) What is meant by significant figures? Write two reasons for using them in
measurements. How to find the uncertainty in a timing experiment such as the
time period of a simple pendulum?
Significant Figures?
Significant figures are the digits in a measurement that are reliably known plus the first
doubtful digit.
�
� Example:
5.63 → 3 significant figures
0.0204 → 3 significant figures (leading zeros not counted)
�
� Two Reasons to Use Significant Figures:
- ✅ They reflect the precision of a measuring instrument.
- ✅ Help in reducing and expressing uncertainty clearly in scientific results.
�
� Uncertainty in Timing Experiment (e.g., Time Period of Pendulum):
To reduce uncertainty in measuring the time period, we:
Measure the time of multiple oscillations (e.g., 20 swings)
Then divide by the number of oscillations
T=Total Time / Number of Oscillations
This averages out reaction errors and gives a more precise value
✅ Uncertainty formula:
ΔT=Least Count/Number of Oscillations
Q1.3 b): The mass of a solid cylinder is 12.85 g, length is 3.35 cm, and diameter is
1.25 cm. Find the density of the material, expressing the uncertainty in the
density.
✅ Step 1: Convert Units to SI
Mass m=12.85 g=0.01285 kg
Length h=3.35 cm=0.0335 m
Diameter d=1.25 cm⇒r=0.625 cm=0.00625 m
✅ Step 2: Find Volume of Cylinder
V=πr^2h=3.14×(0.00625)^2×0.0335≈4.11×10−6 m3
✅ Step 3: Find Density
ρ=m/V=0.01285/4.11×10−6≈3127 kg/m3
✅ Step 4: Expressing Uncertainty (Approximate):
Suppose instrument uncertainties:
Δm=±0.01 g=1×10−5 kg
Δh=±0.01 cm=1×10−4 m
Δr=±0.01 cm=1×10−4 m
Use relative uncertainties:
Δρ / ρ≈Δm / m+2(Δr / r)+Δh / h
Δρ / 3127≈1×10^−5 / 0.01285 + 2×1×10^−4 / 0.00625 + 1×10^−4 / 0.0335
Δρ / 3127≈0.00078+0.032+0.00299≈0.036⇒Δρ≈0.036×3127≈113 kg/m3
�
� Final Answer:
ρ=(3127±113) kg/m3
1.15
Q: Explain with examples the writing of physical quantities into their dimensions. Write its two benefits.
Answer:
Dimensional formula represents a physical quantity using base quantities:
�
� Example:
Force = Mass × Acceleration
[F]=[M][L][T−2]
✅ Benefits:
- Check the correctness of physical equations (homogeneity).
- Helps in unit conversion and deriving relations.
1.16
Q: Check the homogeneity of the relation:
v=√T/m
Where:
v = speed of wave
T = tension (N = kg·m/s²)
m = mass per unit length (kg/m)
Step 1: Dimensions of RHS
T / m=[MLT^−2] / [ML^−1]=[L^2T^−2]
√T/m=[LT^−1]
Step 2: Dimensions of LHS (v):
[v]=[LT−1]
✅ Both sides are equal ⇒ Equation is dimensionally homogeneous