Chapter 1 - Physical Quantities and Measurement Solved 30 Numericals
Strengthen your understanding of Chapter 1: Physical Quantities and Measurement with 30 solved numericals arranged from Easy, Moderate, to HOTS difficulty levels. These step-by-step solutions cover SI units, scientific notation, unit conversion, measuring instruments, significant figures, measurement errors, uncertainty, dimensional analysis, graphs, and practical applications of measurement. Whether you are preparing for school examinations, college assessments, or competitive entrance tests, these numericals will help you develop strong problem-solving skills and build confidence in Physics.
Solved Numerical 1.1
Depth of a Well from Pulley Rotation
Difficulty Level: 🟢 Easy
Problem
A pulley of radius 0.90 m is used to lift a bucket from a well. If the pulley completes 3.6 rotations, determine the depth of the well.
Given
Radius of pulley,
Number of rotations,
Required
Depth of the well, \(d\).
Formula
Distance covered in one complete rotation is:
Therefore,
Solution
Final Answer
Key Concept
The distance moved by a pulley is equal to the number of rotations multiplied by the circumference of the pulley.
Exam Tip 💡
For rotational-distance problems, use \[ \boxed{\text{Distance}=\text{Number of rotations}\times2\pi r} \]
Solved Numerical 1.2
Converting Kilometres into Metres
Difficulty Level: 🟢 Easy
Problem
A road is 7.5 km long. Express its length in metres.
Given
Required
Length in metres.
Formula
Solution
Final Answer
Key Concept
The SI prefix kilo- represents \(10^3\).
Exam Tip 💡
When converting kilometres into metres, multiply the numerical value by 1000.
Solved Numerical 1.3
Converting Mass from Grams to Kilograms
Difficulty Level: 🟢 Easy
Problem
The mass of a laboratory object is 4500 g. Express its mass in kilograms.
Given
Required
Mass in kilograms.
Formula
Solution
Final Answer
Key Concept
The kilogram is the SI base unit of mass.
Exam Tip 💡
Remember: \[ \boxed{1000\,\text{g}=1\,\text{kg}} \]
Solved Numerical 1.4
Writing a Small Measurement in Scientific Notation
Difficulty Level: 🟢 Easy
Problem
Express the length 0.000072 m in scientific notation.
Given
Required
Scientific notation of \(L\).
Formula
Scientific notation is written in the form:
where \(1\leq a<10 p="">
Solution
Moving the decimal point five places to the right gives:
Therefore,
Final Answer
Key Concept
Numbers smaller than 1 are represented by negative powers of 10 in scientific notation.
Exam Tip 💡
If the decimal point moves to the right, the exponent of 10 is negative.
Solved Numerical 1.5
Area of a Rectangular Sheet
Difficulty Level: 🟢 Easy
Problem
A rectangular sheet has a length of 2.5 m and a width of 1.8 m. Calculate its area.
Given
Required
Area of the sheet, \(A\).
Formula
Solution
Final Answer
Key Concept
Area is a derived physical quantity obtained by multiplying two lengths.
Exam Tip 💡
Area is expressed in square units: \[ \boxed{\text{m}\times\text{m}=\text{m}^2} \]
Solved Numerical 1.6
Volume of a Cubical Box
Difficulty Level: 🟢 Easy
Problem
A cubical box has a side length of 0.40 m. Calculate its volume.
Given
Required
Volume of the cube, \(V\).
Formula
Solution
Final Answer
Key Concept
Volume is a three-dimensional physical quantity and is expressed in cubic units.
Exam Tip 💡
For a cube, remember: \[ \boxed{V=a^3} \]
Solved Numerical 1.7
Converting Time into Seconds
Difficulty Level: 🟢 Easy
Problem
A laboratory experiment takes 2 h 25 min. Express the total time in seconds.
Given
Required
Total time in seconds.
Formula
Solution
Time in hours:
Time in minutes:
Total time:
Final Answer
Key Concept
Time is an SI base quantity whose SI base unit is the second.
Exam Tip 💡
Convert every part of a mixed time measurement into the same unit before adding.
Solved Numerical 1.8
Determining the Least Count of a Measuring Instrument
Difficulty Level: 🟢 Easy
Problem
A measuring instrument has a main scale divided into millimetres, with 10 equal subdivisions between two consecutive millimetre marks. Determine the value of each subdivision.
Given
One main-scale division:
Number of subdivisions:
Required
Value of one subdivision (least count).
Formula
Solution
Final Answer
Key Concept
Least count is the smallest measurement that can be reliably read from a measuring instrument.
Exam Tip 💡
A smaller least count generally allows an instrument to measure smaller changes in a quantity.
Solved Numerical 1.9
Calculating Percentage Error in Length Measurement
Difficulty Level: 🟢 Easy
Problem
The actual length of a rod is 50.0 cm, while its measured length is 49.8 cm. Calculate the percentage error.
Given
Actual length:
Measured length:
Required
Percentage error.
Formula
Solution
Absolute error:
Percentage error:
Final Answer
Key Concept
Percentage error compares the magnitude of the measurement error with the actual value.
Exam Tip 💡
Use the absolute value of the error when calculating percentage error.
Solved Numerical 1.10
Calculating Density from Mass and Volume
Difficulty Level: 🟢 Easy
Problem
A metal block has a mass of 540 g and a volume of 200 cm³. Calculate its density.
Given
Required
Density of the metal block, \(\rho\).
Formula
Solution
Converting to SI units:
Final Answer
Key Concept
Density is a derived physical quantity defined as mass per unit volume.
Exam Tip 💡
For density calculations, always check that the units of mass and volume are compatible.
```Solved Numerical 1.11
Adding Measurements with Appropriate Significant Figures
Difficulty Level: 🟡 Moderate
Problem
Calculate the following, giving the answer with the appropriate number of decimal places:
Given
Required
Result with the appropriate number of significant decimal places.
Rule
For addition and subtraction, the final answer should have the same number of decimal places as the quantity having the fewest decimal places.
Solution
The number 5.2 has only one decimal place. Therefore, the answer must be rounded to one decimal place.
Final Answer
Key Concept
In addition and subtraction, rounding is based on the number of decimal places, not the total number of significant figures.
Exam Tip 💡
Always identify the number with the fewest decimal places before rounding an addition or subtraction result.
Solved Numerical 1.12
Multiplying Measurements with Significant Figures
Difficulty Level: 🟡 Moderate
Problem
Calculate the following using the correct number of significant figures:
Given
Required
Product with the appropriate number of significant figures.
Rule
For multiplication and division, the final answer should contain the same number of significant figures as the quantity having the fewest significant figures.
Solution
The number 2.6 has two significant figures. Therefore, the answer should be rounded to two significant figures.
Final Answer
Key Concept
Multiplication and division follow the rule of the fewest significant figures.
Exam Tip 💡
Do not apply the decimal-place rule to multiplication. Use the significant-figure rule instead.
Solved Numerical 1.13
Area of a Sheet with Measurement Uncertainty
Difficulty Level: 🟡 Moderate
Problem
The length and width of a rectangular sheet are measured as
Calculate the area and its absolute uncertainty.
Given
Required
Area \(A\) and absolute uncertainty \(\Delta A\).
Formula
For multiplication, the fractional uncertainties are added:
Solution
First calculate the area:
Now calculate the fractional uncertainty:
Therefore,
Final Answer
Key Concept
For quantities multiplied together, their fractional uncertainties are added.
Exam Tip 💡
When a measured quantity is written with uncertainty, keep the uncertainty and measured value in compatible units.
Solved Numerical 1.14
Calculating Percentage Uncertainty
Difficulty Level: 🟡 Moderate
Problem
The length of a rod is measured as
Calculate its percentage uncertainty.
Given
Required
Percentage uncertainty.
Formula
Solution
Final Answer
Key Concept
Percentage uncertainty expresses the absolute uncertainty as a percentage of the measured value.
Exam Tip 💡
Percentage uncertainty has no physical unit because it is a ratio multiplied by 100.
Solved Numerical 1.15
Converting Speed from Kilometres per Hour to Metres per Second
Difficulty Level: 🟡 Moderate
Problem
Convert 72 km h−1 into metres per second using the conversion-factor method.
Given
Required
Speed in \(\text{m s}^{-1}\).
Conversion Factors
Solution
Final Answer
Key Concept
Conversion factors allow units to be cancelled systematically while keeping the physical quantity unchanged.
Exam Tip 💡
Write conversion factors so that the unwanted units cancel before performing the numerical calculation.
Solved Numerical 1.16
Time Period of a Pendulum with Uncertainty
Difficulty Level: 🟡 Moderate
Problem
The length of a simple pendulum is
and the acceleration due to gravity is
Calculate the time period of the pendulum and its uncertainty.
Given
Required
Time period \(T\) and its uncertainty \(\Delta T\).
Formula
For the fractional uncertainty:
Solution
First calculate the time period:
Now calculate the fractional uncertainty:
Therefore,
Final Answer
Key Concept
For a quantity containing a square root, the fractional uncertainty is multiplied by the corresponding power of one-half.
Exam Tip 💡
For \[ T\propto\sqrt{\frac{l}{g}}, \] the uncertainties in \(l\) and \(g\) contribute only half as strongly to the fractional uncertainty in \(T\).
Solved Numerical 1.17
Dimensional Formula of Force
Difficulty Level: 🟡 Moderate
Problem
Using the equation \(F=ma\), determine the dimensional formula of force.
Given
Dimensions of mass:
Dimensions of acceleration:
Required
Dimensional formula of force, \([F]\).
Formula
Solution
Final Answer
Key Concept
The dimensions of a derived quantity are obtained from the dimensions of the fundamental quantities appearing in its defining equation.
Exam Tip 💡
Remember the fundamental dimensions: \[ \boxed{[M],\ [L],\ [T]} \] for mass, length, and time.
Solved Numerical 1.18
Dimensional Formula of Power
Difficulty Level: 🟡 Moderate
Problem
Determine the dimensional formula of power using the relation
where \(W\) is work and \(t\) is time.
Given
Dimensions of work:
Required
Dimensional formula of power, \([P]\).
Formula
Solution
Final Answer
Key Concept
Power is the rate at which work is done, so its dimensions are obtained by dividing the dimensions of work by time.
Exam Tip 💡
When dividing powers of the same quantity, subtract the exponents.
Solved Numerical 1.19
Verification of the Equation of Motion by Dimensional Analysis
Difficulty Level: 🟡 Moderate
Problem
Verify dimensionally that the equation
is dimensionally correct.
Given
Dimensions of velocity:
Dimensions of acceleration:
Required
Verify the dimensional homogeneity of the equation.
Solution
For the left-hand side:
For the first term on the right-hand side:
For the second term:
Therefore,
All terms have the same dimensions. Hence, the equation is dimensionally homogeneous.
Final Answer
Key Concept
According to the principle of dimensional homogeneity, all terms added or subtracted in a physical equation must have identical dimensions.
Exam Tip 💡
To verify an equation dimensionally, compare the dimensions of both sides rather than comparing their numerical values.
Solved Numerical 1.20
Average Mass from Repeated Measurements
Difficulty Level: 🟡 Moderate
Problem
The mass of an object is measured five times and the readings obtained are:
Calculate the average mass.
Given
Required
Average mass, \(\bar m\).
Formula
Solution
Final Answer
Key Concept
Repeating a measurement and taking the average can reduce the effect of random variations in experimental readings.
Exam Tip 💡
Keep the same appropriate precision in the average as in the original measurements.
Solved Numerical 1.21
Determining the Density of a Cube with Measurement Uncertainty
Difficulty Level: 🔴 HOTS
Problem
A metal cube has a measured side length of
and a mass of
Calculate the density of the metal and its approximate percentage uncertainty.
Given
Required
Density \(\rho\) and its percentage uncertainty.
Formula
For percentage uncertainty:
Solution
First calculate the volume:
Therefore,
Now calculate the fractional uncertainty:
Thus, percentage uncertainty is:
Final Answer
Key Concept
When a quantity is raised to a power, its fractional uncertainty is multiplied by that power. Since \(V=a^3\), the uncertainty in \(a\) contributes three times.
Exam Tip 💡
For \[ Q=x^n, \] the approximate fractional uncertainty is \[ \boxed{\frac{\Delta Q}{Q}=n\frac{\Delta x}{x}}. \]
Solved Numerical 1.22
Finding the Percentage Uncertainty in the Volume of a Sphere
Difficulty Level: 🔴 HOTS
Problem
The radius of a spherical object is measured as
Calculate the percentage uncertainty in its volume.
Given
Required
Percentage uncertainty in volume.
Formula
Therefore,
Solution
Hence,
Final Answer
Key Concept
Because volume depends on the cube of the radius, the percentage uncertainty in the radius is multiplied by three.
Exam Tip 💡
You do not need to calculate the actual volume when only the percentage uncertainty is required.
Solved Numerical 1.23
Determining the Dimensions of Planck's Constant
Difficulty Level: 🔴 HOTS
Problem
Using the equation
where \(E\) is energy and \(f\) is frequency, determine the dimensions of Planck's constant \(h\).
Given
Dimensions of energy:
Dimensions of frequency:
Required
Dimensional formula of \(h\).
Formula
Solution
Final Answer
Key Concept
Dimensional analysis can be used to determine the dimensions of a physical constant from a known physical equation.
Exam Tip 💡
When dividing powers of \(T\), remember: \[ \boxed{T^{-2}\div T^{-1}=T^{-1}} \]
Solved Numerical 1.24
Determining the Dimensions of the Gravitational Constant
Difficulty Level: 🔴 HOTS
Problem
The gravitational force between two masses is given by
Determine the dimensions of the gravitational constant \(G\).
Given
Required
Dimensional formula of \(G\).
Formula
Solution
Final Answer
Key Concept
The dimensions of a constant can be obtained by rearranging its defining physical equation and substituting the dimensions of the known quantities.
Exam Tip 💡
Do not confuse the dimensions of \(G\) with those of acceleration due to gravity \(g\).
Solved Numerical 1.25
Testing a Formula Using Dimensional Analysis
Difficulty Level: 🔴 HOTS
Problem
A student proposes that the time period of a pendulum is given by
Use dimensional analysis to determine whether the proposed equation is dimensionally correct.
Given
Dimensions of length:
Dimensions of acceleration due to gravity:
Required
Verify the dimensions of the equation.
Solution
Consider the right-hand side:
The factor \(2\pi\) is dimensionless. Therefore,
The left-hand side also has dimensions:
Both sides have the same dimensions.
Final Answer
Key Concept
Dimensional homogeneity requires both sides of a physical equation to have identical dimensions.
Exam Tip 💡
Dimensional analysis can verify dimensional consistency, but it cannot prove that a numerical constant such as \(2\pi\) is correct.
Solved Numerical 1.26
Comparing Accuracy and Precision of Measurements
Difficulty Level: 🔴 HOTS
Problem
The accepted value of a length is 10.00 cm. Two students obtain the following repeated measurements:
Which student demonstrates greater accuracy and which demonstrates greater precision?
Given
Required
Compare the accuracy and precision of the two sets of measurements.
Solution
Student A's readings are very close to the accepted value of \(10.00\,\text{cm}\) and are also close to one another.
Student B's readings are very close to one another, but they are significantly different from the accepted value.
Therefore, Student A has both high accuracy and high precision. Student B has high precision but lower accuracy.
Final Answer
Key Concept
Accuracy describes closeness to the accepted value, whereas precision describes the closeness of repeated measurements to one another.
Exam Tip 💡
A set of readings can be precise without being accurate.
Solved Numerical 1.27
Determining the Number of Significant Figures in a Measurement
Difficulty Level: 🔴 HOTS
Problem
A measurement is reported as
Determine the number of significant figures and express the measurement in scientific notation without changing its precision.
Given
Required
Number of significant figures and scientific notation.
Solution
The zeros before the first non-zero digit are not significant. The zero between non-zero digits is significant, and the trailing zeros after the decimal point are also significant.
Therefore, the significant digits are:
Thus, there are five significant figures.
Moving the decimal point three places to the right gives:
Final Answer
Key Concept
Scientific notation makes the number of significant figures explicit.
Exam Tip 💡
Zeros between non-zero digits are significant, while zeros before the first non-zero digit are not significant.
Solved Numerical 1.28
Determining the Least Count of a Vernier Calipers
Difficulty Level: 🔴 HOTS
Problem
The main scale of a vernier calipers has 1 mm as its smallest division. Ten vernier-scale divisions are equal to nine main-scale divisions. Determine the least count of the vernier calipers.
Given
Required
Least count of the vernier calipers.
Formula
First determine one vernier-scale division:
The least count is:
Solution
Therefore,
Final Answer
Key Concept
The least count of a vernier instrument is the difference between one main-scale division and one vernier-scale division.
Exam Tip 💡
Always determine the value of one VSD before calculating the least count.
Solved Numerical 1.29
Combining Uncertainties in a Derived Quantity
Difficulty Level: 🔴 HOTS
Problem
The mass of an object is measured as
and its volume is measured as
Calculate the density and its percentage uncertainty.
Given
Required
Density \(\rho\) and percentage uncertainty.
Formula
For division, fractional uncertainties are added:
Solution
Calculate the density:
Calculate fractional uncertainty:
Therefore, percentage uncertainty is:
The approximate absolute uncertainty is:
Final Answer
Key Concept
For a quotient, the fractional uncertainties of the numerator and denominator are added.
Exam Tip 💡
For multiplication or division: \[ \boxed{\frac{\Delta Q}{Q} = \frac{\Delta A}{A} + \frac{\Delta B}{B}} \]
Solved Numerical 1.30
Finding an Unknown Physical Quantity from Dimensional Reasoning
Difficulty Level: 🔴 HOTS
Problem
Suppose the speed \(v\) of a wave depends on its frequency \(f\) and wavelength \(\lambda\) according to
where \(k\) is a dimensionless constant. Use dimensional analysis to determine the values of \(a\) and \(b\).
Given
The dimensions are:
Required
Values of \(a\) and \(b\).
Solution
Taking dimensions of both sides:
Therefore:
Comparing powers of \(L\):
Comparing powers of \(T\):
Thus,
Since \(k\) is dimensionless, dimensional analysis alone cannot determine its numerical value. For the known wave relation, \(k=1\).
Final Answer
Hence, the dimensional form is:
Key Concept
Dimensional analysis can be used to determine the powers of physical quantities in a proposed relationship.
Exam Tip 💡
Dimensional analysis can determine powers and dimensions, but it generally cannot determine dimensionless numerical constants.
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