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Fundamental and Derived Quantities

20 questions

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Question 1Question →

Which of the following physical quantities is classified as a fundamental quantity in the International System of Units (SI)?

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Answer: Electric current

Answer

Electric current
Electric current is one of the seven fundamental physical quantities defined by the International System of Units (SI). It serves as a base quantity from which other electrical quantities (such as charge and potential difference) are derived.

Step-by-Step Solution

1
Identify the definition of a fundamental quantity
Fundamental quantities are independent physical quantities that cannot be expressed in terms of other quantities.
SI defines seven basic quantities: length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.
2
Evaluate the options against the list of SI fundamental quantities
Electric current is a basic quantity (measured in amperes, AA), whereas force, velocity, and density are derived from mass, length, and time.
Only electric current is an independent base quantity.

Key Concept

Fundamental quantities are basic physical quantities that are independent of other quantities.
Question 2Question →

A physical quantity XX is defined by the expression X=P⋅Vm⋅tX = \frac{P \cdot V}{m \cdot t}, where PP represents pressure, VV represents volume, mm represents mass, and tt represents time. Which of the following statements correctly classifies XX and expresses its unit strictly in terms of fundamental SI base units?

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Answer: XX is a derived quantity, and its unit in SI base units is m2⋅s−3\text{m}^2 \cdot \text{s}^{-3}.

Answer

The physical quantity XX is a derived quantity, and its unit expressed strictly in fundamental SI base units is m2⋅s−3\text{m}^2 \cdot \text{s}^{-3}.
The quantity XX is defined via a mathematical formula involving pressure, volume, mass, and time, which classifies it as a derived quantity. Replacing each component with its fundamental SI base units gives pressure as kg⋅m−1⋅s−2\text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-2} and volume as m3\text{m}^3, making the numerator kg⋅m2⋅s−2\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}. Dividing by the denominator (m⋅t=kg⋅sm \cdot t = \text{kg} \cdot \text{s}) cancels out kilograms and leaves m2⋅s−3\text{m}^2 \cdot \text{s}^{-3}, consisting purely of fundamental base units.

Step-by-Step Solution

1
Classify the physical quantity XX
XX is a derived physical quantity.
Fundamental physical quantities in the SI system are length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity. Because XX is calculated from a combination of other quantities, it is derived.
2
Express pressure (PP) and volume (VV) in terms of fundamental SI base units
P=kg⋅m−1⋅s−2P = \text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-2} and V=m3V = \text{m}^3.
Pressure is defined as force per unit area (kg⋅m⋅s−2m2)\left(\frac{\text{kg} \cdot \text{m} \cdot \text{s}^{-2}}{\text{m}^2}\right), and volume has the base unit m3\text{m}^3.
3
Calculate the base unit expression for the numerator P⋅VP \cdot V
P⋅V=(kg⋅m−1⋅s−2)⋅(m3)=kg⋅m2⋅s−2P \cdot V = (\text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-2}) \cdot (\text{m}^3) = \text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}.
Combining the powers of length (metres) gives −1+3=2-1 + 3 = 2.
4
Divide by the denominator m⋅tm \cdot t to obtain the base units of XX
X=kg⋅m2⋅s−2kg⋅s=m2⋅s−3X = \frac{\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}}{\text{kg} \cdot \text{s}} = \text{m}^2 \cdot \text{s}^{-3}.
The unit of mass (kg\text{kg}) cancels completely, and dividing by time (s\text{s}) reduces the exponent of seconds from −2-2 to −3-3.

Key Concept

Fundamental quantities are independent base quantities defined by the SI system, whereas derived quantities are defined algebraically from fundamental quantities. Reducing derived units to SI base units requires breaking down all non-base units into metres (m), kilograms (kg), seconds (s), amperes (A), kelvins (K), moles (mol), or candelas (cd).
Estimated Time:2m 0s
Question 3Question →

Which of the following groups consists exclusively of fundamental physical quantities?

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Answer: Mass, thermodynamic temperature, and luminous intensity

Answer

The group containing mass, thermodynamic temperature, and luminous intensity consists exclusively of fundamental physical quantities.
Mass, thermodynamic temperature, and luminous intensity are three of the seven internationally recognized SI base (fundamental) physical quantities.

Step-by-Step Solution

1
Identify the seven fundamental SI physical quantities
The seven base quantities are length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.
Fundamental quantities are independent quantities that cannot be defined in terms of other physical quantities.
2
Evaluate each provided option against the list of base quantities
Electric charge, force, and weight are derived quantities. Only mass, thermodynamic temperature, and luminous intensity are all fundamental.
Any quantity derived by multiplying or dividing base quantities is derived.

Key Concept

Fundamental quantities are basic physical quantities that are independent of one another and form the basis from which derived quantities are obtained.
Question 4Question →

A physical quantity ZZ is defined as the ratio of the product of impulse and linear velocity to the product of electric current and electric potential difference. When ZZ is fully resolved into fundamental physical quantities, which fundamental quantity does ZZ represent?

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Answer: Time

Answer

Time
The correct answer is Time because reducing both the numerator (energy) and denominator (electrical power) to base fundamental dimensions yields ML2T−2ML2T−3=T\frac{M L^2 T^{-2}}{M L^2 T^{-3}} = T, which corresponds directly to the fundamental quantity of Time.

Step-by-Step Solution

1
Express impulse and velocity in terms of fundamental base quantities (Mass MM, Length LL, Time TT)
Impulse=Force×Time=MLT−1\text{Impulse} = \text{Force} \times \text{Time} = M L T^{-1}. Velocity=LT−1\text{Velocity} = L T^{-1}. Product =(MLT−1)(LT−1)=ML2T−2= (M L T^{-1})(L T^{-1}) = M L^2 T^{-2}.
To evaluate the numerator in fundamental base dimensions.
2
Express electric current and potential difference in terms of fundamental base quantities
Current=I\text{Current} = I. Potential Difference=WorkCharge=ML2T−2IT=ML2T−3I−1\text{Potential Difference} = \frac{\text{Work}}{\text{Charge}} = \frac{M L^2 T^{-2}}{I T} = M L^2 T^{-3} I^{-1}. Product =I×(ML2T−3I−1)=ML2T−3= I \times (M L^2 T^{-3} I^{-1}) = M L^2 T^{-3}.
To evaluate the denominator in fundamental base dimensions.
3
Divide the numerator by the denominator to simplify quantity ZZ
Z=ML2T−2ML2T−3=M1−1L2−2T−2−(−3)=T1Z = \frac{M L^2 T^{-2}}{M L^2 T^{-3}} = M^{1-1} L^{2-2} T^{-2 - (-3)} = T^1.
Determining the net fundamental quantity after all derived units cancel out.

Key Concept

Fundamental and Derived Quantities
Question 5Question →

Match each physical quantity listed on the left with its correct SI classification or fundamental unit decomposition on the right.

Click a left item, then click its matching right item

Items

Electric current
Pressure
Thermodynamic temperature
Impulse

Matches

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Answer

Electric current corresponds to the fundamental quantity measured in amperes (A\text{A}); Pressure corresponds to the derived quantity with base SI units kg⋅m−1⋅s−2\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}; Thermodynamic temperature corresponds to the fundamental quantity measured in kelvin (K\text{K}); Impulse corresponds to the derived quantity with base SI units kg⋅m⋅s−1\text{kg}\cdot\text{m}\cdot\text{s}^{-1}.
Electric current and thermodynamic temperature are fundamental SI quantities defined independently with units ampere (A\text{A}) and kelvin (K\text{K}). Pressure and impulse are derived quantities whose definitions rely on fundamental quantities, breaking down into kg⋅m−1⋅s−2\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2} and kg⋅m⋅s−1\text{kg}\cdot\text{m}\cdot\text{s}^{-1} respectively.

Step-by-Step Solution

1
Identify fundamental physical quantities
Electric current and thermodynamic temperature are fundamental quantities with SI base units ampere (A\text{A}) and kelvin (K\text{K}) respectively.
Fundamental quantities are basic physical quantities that do not depend on any other physical quantity for their definition.
2
Decompose pressure into base SI units
Pressure=ForceArea=kg⋅m⋅s−2m2=kg⋅m−1⋅s−2\text{Pressure} = \frac{\text{Force}}{\text{Area}} = \frac{\text{kg}\cdot\text{m}\cdot\text{s}^{-2}}{\text{m}^2} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}.
Derived quantities must be reduced to combinations of mass (kg\text{kg}), length (m\text{m}), and time (s\text{s}).
3
Decompose impulse into base SI units
Impulse=Force×Time=(kg⋅m⋅s−2)×s=kg⋅m⋅s−1\text{Impulse} = \text{Force} \times \text{Time} = (\text{kg}\cdot\text{m}\cdot\text{s}^{-2}) \times \text{s} = \text{kg}\cdot\text{m}\cdot\text{s}^{-1}.
Impulse is defined as change in momentum or force applied over a time interval.

Key Concept

Classification of fundamental quantities versus derived quantities and resolution of derived units into fundamental SI units.
Question 6Question →

Match each physical quantity on the left with its correct fundamental SI base unit expression or fundamental status on the right.

Click a left item, then click its matching right item

Items

Thermodynamic temperature
Electric charge
Linear momentum
Power

Matches

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Answer

Thermodynamic temperature matches with 'Fundamental physical quantity measured in kelvin (K)'; Electric charge matches with 'Derived physical quantity expressed in base SI units as A·s'; Linear momentum matches with 'Derived physical quantity expressed in base SI units as kg·m·s⁻¹'; Power matches with 'Derived physical quantity expressed in base SI units as kg·m²·s⁻³'.
Each physical quantity is correctly paired with either its fundamental status or its base SI unit breakdown derived from core physics definitions.

Step-by-Step Solution

1
Identify fundamental quantities versus derived quantities.
Thermodynamic temperature is a basic fundamental quantity (unit: K\text{K}). Electric current is fundamental (unit: A\text{A}), but electric charge is derived (Q=ItQ = I t).
Fundamental quantities cannot be defined in terms of other physical quantities.
2
Decompose Electric Charge into base units.
Since Q=I⋅tQ = I \cdot t, its unit is A⋅s\text{A}\cdot\text{s}.
Current is measured in amperes and time in seconds.
3
Decompose Linear Momentum into base units.
p=m⋅v⇒unit=kg⋅m⋅s−1p = m \cdot v \Rightarrow \text{unit} = \text{kg} \cdot \text{m}\cdot\text{s}^{-1}.
Mass is in kilograms and velocity is in meters per second.
4
Decompose Power into base units.
P=Wt=F⋅dt=(m⋅a)⋅dt⇒kg⋅(m⋅s−2)⋅ms=kg⋅m2⋅s−3P = \frac{W}{t} = \frac{F \cdot d}{t} = \frac{(m \cdot a) \cdot d}{t} \Rightarrow \frac{\text{kg} \cdot (\text{m}\cdot\text{s}^{-2}) \cdot \text{m}}{\text{s}} = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}.
Power is work done per unit time.

Key Concept

Fundamental and Derived Quantities
Question 7Question →

A laboratory technician records four physical quantities during an experiment: electric current, thermodynamic temperature, electric charge, and luminous intensity. Which of these recorded quantities is a derived physical quantity?

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Answer: Electric charge

Answer

Electric charge is the derived physical quantity.
Electric charge is a derived physical quantity because it is defined mathematically in terms of fundamental quantities: electric charge equals electric current multiplied by time (Q=I⋅tQ = I \cdot t). Its SI unit, the coulomb (C), is defined as an ampere-second (1 C=1 A⋅s1\text{ C} = 1\text{ A}\cdot\text{s}).

Step-by-Step Solution

1
Identify the seven fundamental physical quantities defined in the SI system.
The seven base quantities are length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.
Fundamental quantities are independent base quantities from which all other physical quantities are derived.
2
Analyze each quantity given in the scenario against the list of fundamental quantities.
Electric current, thermodynamic temperature, and luminous intensity are fundamental quantities. Electric charge is not on the list of base quantities.
Electric charge (QQ) is derived from the base quantities electric current (II) and time (tt) through the relation Q=I⋅tQ = I \cdot t.

Key Concept

Fundamental quantities are basic physical quantities that do not depend on other quantities. Derived quantities are defined in terms of fundamental quantities.
Question 8Question →

Match each physical quantity listed on the left with its corresponding SI classification and base unit representation on the right.

Click a left item, then click its matching right item

Items

Electric current
Thermodynamic temperature
Electric potential difference
Specific heat capacity

Matches

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Answer

Electric current matches Fundamental quantity measured in amperes; Thermodynamic temperature matches Fundamental quantity measured in kelvins; Electric potential difference matches Derived quantity expressed as kg·m²·s⁻³·A⁻¹; Specific heat capacity matches Derived quantity expressed as m²·s⁻²·K⁻¹.
Electric current and thermodynamic temperature are basic SI fundamental quantities. Electric potential difference and specific heat capacity are derived quantities whose fundamental unit decompositions follow directly from their governing formulas.

Step-by-Step Solution

1
Identify fundamental physical quantities and their base units
Electric current and thermodynamic temperature are fundamental SI quantities with base units ampere (A\text{A}) and kelvin (K\text{K}) respectively.
Fundamental physical quantities are defined independently and serve as the foundation for the SI system.
2
Decompose electric potential difference into SI base units
Electric potential difference V=WorkCharge=kg⋅m2⋅s−2A⋅s=kg⋅m2⋅s−3⋅A−1V = \frac{\text{Work}}{\text{Charge}} = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{A}\cdot\text{s}} = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}.
Because it is expressed by combining fundamental quantities, it is a derived quantity.
3
Decompose specific heat capacity into SI base units
Specific heat capacity c=EnergyMass×Temperature change=kg⋅m2⋅s−2kg⋅K=m2⋅s−2⋅K−1c = \frac{\text{Energy}}{\text{Mass} \times \text{Temperature change}} = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{kg}\cdot\text{K}} = \text{m}^2\cdot\text{s}^{-2}\cdot\text{K}^{-1}.
It is calculated from energy, mass, and temperature, making it a derived quantity.

Key Concept

Fundamental quantities are independent basic quantities, whereas derived quantities are formed through algebraic combination of fundamental quantities.
Estimated Time:1m 30s
Question 9Question →

Match each physical quantity to its correct classification as either a fundamental or a derived physical quantity.

Click a left item, then click its matching right item

Items

Luminous intensity
Mass
Force
Electric potential

Matches

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Answer

Luminous intensity matches with fundamental quantity measuring light brightness; Mass matches with fundamental quantity measuring quantity of matter; Force matches with derived quantity defined as rate of change of linear momentum; Electric potential matches with derived quantity defined as work done per unit electric charge.
Luminous intensity and mass are two of the seven base SI quantities. Force and electric potential are derived quantities defined through mathematical combinations of base quantities.

Step-by-Step Solution

1
Identify the fundamental physical quantities
Luminous intensity and Mass are fundamental physical quantities.
Fundamental physical quantities are basic quantities that do not depend on any other physical quantity for their definition.
2
Identify the derived physical quantities
Force and Electric potential are derived physical quantities.
Derived physical quantities are obtained by combining fundamental physical quantities through mathematical relationships.

Key Concept

Fundamental quantities (length, mass, time, electric current, thermodynamic temperature, amount of substance, luminous intensity) are independent, whereas derived quantities are formed by combining fundamental quantities.
Question 10Question →

Consider the following four physical quantities:
I. Electric current
II. Electric charge
III. Thermodynamic temperature
IV. Heat capacity

Which of the following pairs correctly identifies a fundamental physical quantity followed by a derived physical quantity?

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Answer: Thermodynamic temperature and Heat capacity

Answer

Thermodynamic temperature and Heat capacity
Thermodynamic temperature is one of the seven base SI fundamental quantities. Heat capacity is a derived quantity defined as thermal energy per temperature change (J/KJ/K). Thus, the pair of Thermodynamic temperature and Heat capacity strictly follows the order of a fundamental quantity followed by a derived quantity.

Step-by-Step Solution

1
Identify the fundamental physical quantities among the given list.
Electric current (I) and Thermodynamic temperature (III) are base SI quantities, so they are fundamental.
Fundamental quantities are independent physical quantities that cannot be defined in terms of other quantities.
2
Identify the derived physical quantities among the given list.
Electric charge (II, defined by Q=I⋅tQ = I \cdot t) and Heat capacity (IV, defined by C=QΔTC = \frac{Q}{\Delta T}) are derived quantities.
Derived quantities are defined mathematically from combinations of fundamental physical quantities.
3
Select the option that lists a fundamental quantity followed by a derived quantity.
Thermodynamic temperature (fundamental) and Heat capacity (derived) matches the required order.
Thermodynamic temperature is fundamental, and heat capacity is derived.

Key Concept

Classification of physical quantities into base (fundamental) and derived quantities in the SI system.
Question 11Question →

Match each physical quantity to its correct physical quantity classification and corresponding SI base unit expression.

Click a left item, then click its matching right item

Items

Luminous intensity
Electric potential
Specific heat capacity
Thermodynamic temperature

Matches

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Answer

Luminous intensity matches with fundamental quantity in cd\text{cd}; Electric potential matches with derived quantity in kg⋅m2⋅s−3⋅A−1\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}; Specific heat capacity matches with derived quantity in m2⋅s−2⋅K−1\text{m}^2\cdot\text{s}^{-2}\cdot\text{K}^{-1}; Thermodynamic temperature matches with fundamental quantity in K\text{K}.
Luminous intensity and thermodynamic temperature are fundamental SI quantities with base units cd and K. Electric potential and specific heat capacity are derived quantities whose definitions reduce to kg⋅m2⋅s−3⋅A−1\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1} and m2⋅s−2⋅K−1\text{m}^2\cdot\text{s}^{-2}\cdot\text{K}^{-1} respectively.

Step-by-Step Solution

1
Identify fundamental physical quantities
Luminous intensity and thermodynamic temperature are base SI quantities measured in candela (cd) and kelvin (K) respectively.
Base quantities cannot be defined in terms of other physical quantities.
2
Decompose derived quantities into base SI units
Electric potential V=WqV = \frac{W}{q} yields kg⋅m2⋅s−3⋅A−1\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}. Specific heat capacity c=QmΔTc = \frac{Q}{m\Delta T} yields m2⋅s−2⋅K−1\text{m}^2\cdot\text{s}^{-2}\cdot\text{K}^{-1}.
Derived quantities originate from mathematical combinations of fundamental quantities.

Key Concept

Classification of physical quantities into fundamental (base) and derived categories and resolution into SI base units
Question 12Question →

Match each composite physical quantity or ratio on the left with its correct fundamental (SI base) unit decomposition on the right.

Click a left item, then click its matching right item

Items

Electric potential gradient
Coefficient of dynamic viscosity
Ratio of Planck's constant to moment of inertia
Specific latent heat divided by spatial temperature gradient

Matches

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Answer

Electric potential gradient matches kg⋅m⋅s−3⋅A−1\text{kg}\cdot\text{m}\cdot\text{s}^{-3}\cdot\text{A}^{-1}; Coefficient of dynamic viscosity matches kg⋅m−1⋅s−1\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-1}; Ratio of Planck's constant to moment of inertia matches s−1\text{s}^{-1}; Specific latent heat divided by spatial temperature gradient matches m3⋅s−2⋅K−1\text{m}^3\cdot\text{s}^{-2}\cdot\text{K}^{-1}.
Each physical quantity or ratio is systematically reduced to its SI base quantities (mass in kg, length in m, time in s, electric current in A, thermodynamic temperature in K) by substituting fundamental definitions of derived units.

Step-by-Step Solution

1
Decompose electric potential gradient into fundamental SI base units
Electric potential gradient=Electric PotentialDistance=WorkCharge×Distance=kg⋅m2⋅s−2A⋅s×m=kg⋅m⋅s−3⋅A−1\text{Electric potential gradient} = \frac{\text{Electric Potential}}{\text{Distance}} = \frac{\text{Work}}{\text{Charge} \times \text{Distance}} = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{A}\cdot\text{s} \times \text{m}} = \text{kg}\cdot\text{m}\cdot\text{s}^{-3}\cdot\text{A}^{-1}.
Electric potential is defined as energy per unit charge, and potential gradient is its spatial rate of change.
2
Decompose coefficient of dynamic viscosity into fundamental SI base units
η=Force×DistanceArea×Velocity=(kg⋅m⋅s−2)×mm2×(m⋅s−1)=kg⋅m−1⋅s−1\eta = \frac{\text{Force} \times \text{Distance}}{\text{Area} \times \text{Velocity}} = \frac{(\text{kg}\cdot\text{m}\cdot\text{s}^{-2}) \times \text{m}}{\text{m}^2 \times (\text{m}\cdot\text{s}^{-1})} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-1}.
Newton's law of viscosity relates shear force to surface area and velocity gradient.
3
Determine the base unit ratio of Planck's constant to moment of inertia
\frac{h}{I} = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-1}}{\text{kg}\cdot\text{m}^2} = \text{s}^{-1}.
Planck's constant carries dimensions of angular momentum, while moment of inertia is mass multiplied by distance squared.
4
Decompose specific latent heat divided by spatial temperature gradient
\frac{L}{\frac{\Delta T}{\Delta x}} = \frac{\text{J}\cdot\text{kg}^{-1}}{\text{K}\cdot\text{m}^{-1}} = \frac{\text{m}^2\cdot\text{s}^{-2}}{\text{K}\cdot\text{m}^{-1}} = \text{m}^3\cdot\text{s}^{-2}\cdot\text{K}^{-1}.
Specific heat quantities represent thermal energy per unit mass, whereas temperature gradient represents thermal variation per unit displacement.

Key Concept

Fundamental SI base unit decomposition of derived physical quantities
Question 13Question →

A physical quantity ZZ is defined by the expression Z=P⋅VI⋅tZ = \frac{P \cdot V}{I \cdot t}, where PP represents pressure, VV represents volume, II represents electric current, and tt represents time. Which of the following statements correctly classifies quantity ZZ and expresses it in fundamental SI base units?

Show answer & explanation

Answer: Quantity ZZ is a derived quantity with fundamental SI base units of kg⋅m2⋅s−3⋅A−1\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}.

Answer

Quantity ZZ is a derived quantity with fundamental SI base units of kg⋅m2⋅s−3⋅A−1\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}.
The quantity ZZ represents electric potential (voltage), which is defined as work done per unit electric charge (W/QW/Q). Since electric potential is derived from mass, length, time, and electric current, it is a derived physical quantity. Expanding P⋅VP \cdot V yields energy units (kg⋅m2⋅s−2\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}), and dividing by I⋅tI \cdot t (A⋅s\text{A}\cdot\text{s}) gives the correct base units of kg⋅m2⋅s−3⋅A−1\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}.

Step-by-Step Solution

1
Determine the SI base unit decomposition for pressure (PP) and volume (VV).
Pressure P=ForceArea=kg⋅m⋅s−2m2=kg⋅m−1⋅s−2P = \frac{\text{Force}}{\text{Area}} = \frac{\text{kg}\cdot\text{m}\cdot\text{s}^{-2}}{\text{m}^2} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}. Volume V=m3V = \text{m}^3. Therefore, P⋅V=(kg⋅m−1⋅s−2)⋅m3=kg⋅m2⋅s−2P \cdot V = (\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}) \cdot \text{m}^3 = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}.
Decomposing composite physical quantities into base units of mass, length, and time.
2
Determine the SI base unit decomposition for the denominator I⋅tI \cdot t.
I⋅t=Electric current×Time=A⋅sI \cdot t = \text{Electric current} \times \text{Time} = \text{A}\cdot\text{s}.
Electric current (amperes, A) and time (seconds, s) are fundamental SI quantities.
3
Evaluate the full expression for Z=P⋅VI⋅tZ = \frac{P \cdot V}{I \cdot t} in SI base units.
Base units of Z=kg⋅m2⋅s−2A⋅s=kg⋅m2⋅s−3⋅A−1Z = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{A}\cdot\text{s}} = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}.
Apply laws of indices to simplify base unit ratios.
4
Classify physical quantity ZZ.
Because ZZ (electric potential) is defined in terms of fundamental quantities (kg, m, s, A) rather than existing independently, it is a derived quantity.
The seven fundamental SI quantities are length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.

Key Concept

Classification of physical quantities into fundamental and derived types and resolving complex derived quantities into base SI units.
Question 14Question →

The rate of change of linear momentum per unit cross-sectional area is a derived physical quantity. When resolved into fundamental SI base units, which of the following physical quantities has the exact same SI base unit decomposition?

Show answer & explanation

Answer: Energy density

Answer

Energy density
The rate of change of momentum is force (FF), which has SI base units of kg⋅m⋅s−2\text{kg}\cdot\text{m}\cdot\text{s}^{-2}. Dividing by area (m2\text{m}^2) gives kg⋅m−1⋅s−2\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}. Energy density is defined as energy per unit volume, which decomposes to kg⋅m2⋅s−2m3=kg⋅m−1⋅s−2\frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{m}^3} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}. Both derived quantities possess identical fundamental SI base unit representations.

Step-by-Step Solution

1
Determine the physical definition of the rate of change of linear momentum per unit area.
By Newton's second law, rate of change of momentum equals force (F=ΔpΔtF = \frac{\Delta p}{\Delta t}). Dividing by cross-sectional area (AA) yields force per unit area (pressure or stress), FA\frac{F}{A}.
Force is defined as the time rate of change of linear momentum.
2
Express force per unit area in terms of fundamental SI base units.
Force=mass×acceleration=kg⋅m⋅s−2\text{Force} = \text{mass} \times \text{acceleration} = \text{kg}\cdot\text{m}\cdot\text{s}^{-2}. Therefore, FA=kg⋅m⋅s−2m2=kg⋅m−1⋅s−2\frac{F}{A} = \frac{\text{kg}\cdot\text{m}\cdot\text{s}^{-2}}{\text{m}^2} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}.
Mass (kg\text{kg}), length (m\text{m}), and time (s\text{s}) are fundamental base quantities.
3
Analyze energy density in fundamental SI base units.
Energy density=EnergyVolume=kg⋅m2⋅s−2m3=kg⋅m−1⋅s−2\text{Energy density} = \frac{\text{Energy}}{\text{Volume}} = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{m}^3} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}.
Decomposing energy into work (force ×\times distance) and dividing by volume reveals its base unit equivalence.

Key Concept

Decomposition of derived quantities into fundamental SI base units
Estimated Time:1m 30s
Question 15Question →

During an investigation on electrical circuits, a student measures electric current, potential difference, time, and thermodynamic temperature. Which of the measured physical quantities is a derived quantity?

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Answer: Potential difference

Answer

Potential difference
Potential difference is a derived physical quantity because it is derived from work (energy) and electric charge (V=WQ=WI⋅tV = \frac{W}{Q} = \frac{W}{I \cdot t}), unlike electric current, time, and thermodynamic temperature which are fundamental SI quantities.

Step-by-Step Solution

1
Identify fundamental (base) physical quantities
The seven SI fundamental quantities are mass, length, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.
Fundamental quantities are basic quantities that do not depend on any other physical quantity.
2
Classify the given quantities
Electric current, time, and thermodynamic temperature are base quantities. Potential difference (V=WQV = \frac{W}{Q}) is derived from work and electric charge.
Derived quantities are physical quantities defined by combining fundamental quantities.

Key Concept

Fundamental and Derived Quantities
Estimated Time:1m 0s
Question 16Question →

Which of the following combinations consists entirely of derived physical quantities?

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Answer: Volume, momentum, and electrical potential difference

Answer

The combination comprising volume, momentum, and electrical potential difference consists entirely of derived physical quantities.
The combination containing volume, momentum, and electrical potential difference consists entirely of derived physical quantities because volume depends on length (L3L^3), momentum depends on mass, length, and time (M⋅L⋅T−1M\cdot L\cdot T^{-1}), and electrical potential difference depends on mass, length, time, and electric current (M⋅L2⋅T−3⋅I−1M\cdot L^2\cdot T^{-3}\cdot I^{-1}). None of these three are base/fundamental quantities.

Step-by-Step Solution

1
Identify the seven fundamental physical quantities in SI units.
The seven fundamental quantities are mass, length, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.
Fundamental quantities serve as the basic foundation from which all other physical quantities are defined.
2
Classify each physical quantity present in the given combinations.
Volume (m3m^3), momentum (kg⋅m/skg\cdot m/s), and potential difference (VV or kg⋅m2/(A⋅s3)kg\cdot m^2/(A\cdot s^3)) are all derived from fundamental units. Other combinations contain fundamental quantities such as length, mass, time, temperature, or electric current.
Derived quantities are physical quantities defined by mathematical combinations of fundamental quantities.
3
Select the option where every listed quantity is a derived quantity.
The group containing volume, momentum, and electrical potential difference is the only set where all items are derived quantities.
It fulfills the requirement of containing exclusively derived physical quantities.

Key Concept

Fundamental quantities are independent quantities defined by international standard protocols, whereas derived quantities are formed by mathematical combinations of fundamental quantities.
Question 17Question →

Match each of the physical quantities given on the left with its corresponding fundamental SI status or base unit resolution on the right.

Click a left item, then click its matching right item

Items

Luminous intensity
Linear momentum
Thermodynamic temperature
Specific heat capacity

Matches

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Answer

Luminous intensity matches with Fundamental quantity measured in candela (cd); Linear momentum matches with Derived quantity expressed in kg·m·s⁻¹; Thermodynamic temperature matches with Fundamental quantity measured in kelvin (K); Specific heat capacity matches with Derived quantity expressed in m²·s⁻²·K⁻¹.
Luminous intensity and thermodynamic temperature are fundamental physical quantities with SI units candela (cd\text{cd}) and kelvin (K\text{K}) respectively. Linear momentum (p=mvp = mv) and specific heat capacity (c=QmΔTc = \frac{Q}{m \Delta T}) are derived physical quantities whose fundamental SI base unit resolutions are kg⋅m⋅s−1\text{kg}\cdot\text{m}\cdot\text{s}^{-1} and m2⋅s−2⋅K−1\text{m}^2\cdot\text{s}^{-2}\cdot\text{K}^{-1} respectively.

Step-by-Step Solution

1
Identify fundamental physical quantities and their base SI units
Luminous intensity (measured in cd\text{cd}) and thermodynamic temperature (measured in K\text{K}) are base physical quantities that cannot be expressed in terms of other quantities.
The standard SI system establishes 7 base independent quantities.
2
Resolve derived quantities into fundamental SI base units
Linear momentum (p=mvp = mv) has units kg⋅m⋅s−1\text{kg}\cdot\text{m}\cdot\text{s}^{-1}. Specific heat capacity (c=QmΔTc = \frac{Q}{m\Delta T}) has units kg⋅m2⋅s−2kg⋅K=m2⋅s−2⋅K−1\frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{kg}\cdot\text{K}} = \text{m}^2\cdot\text{s}^{-2}\cdot\text{K}^{-1}.
Derived physical quantities are formed by combining fundamental quantities algebraically according to physical laws.
3
Match each physical quantity to its correct description
Luminous intensity →\rightarrow candela (cd\text{cd}); Linear momentum →\rightarrow kg⋅m⋅s−1\text{kg}\cdot\text{m}\cdot\text{s}^{-1}; Thermodynamic temperature →\rightarrow kelvin (K\text{K}); Specific heat capacity →\rightarrow m2⋅s−2⋅K−1\text{m}^2\cdot\text{s}^{-2}\cdot\text{K}^{-1}.
Each pair directly aligns the physical quantity with its base classification and unit derivation.

Key Concept

Classification of fundamental and derived physical quantities and their resolution into SI base units
Question 18Question →

A student investigating the physical properties of a uniform metallic rod records its mass, length, thermodynamic temperature, and mass density. Which of the recorded physical quantities is classified as a derived quantity?

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Answer: Mass density

Answer

Mass density is the derived physical quantity.
Mass density is defined as mass per unit volume (ρ=mV \rho = \frac{m}{V} ). Because it is obtained by combining the fundamental quantities of mass and length, it is a derived physical quantity.

Step-by-Step Solution

1
Identify the seven fundamental SI physical quantities
The seven base quantities are length, mass, time, electric current, thermodynamic temperature, luminous intensity, and amount of substance.
Fundamental quantities are independent physical quantities that cannot be defined in terms of other physical quantities.
2
Classify each physical quantity given in the scenario
Mass, length, and thermodynamic temperature are fundamental quantities. Mass density is defined as mass divided by volume (ρ=mV \rho = \frac{m}{V} ).
Derived quantities are physical quantities obtained by mathematical combinations of fundamental quantities.

Key Concept

Classification of Fundamental and Derived Quantities
Question 19Question →

Match each physical quantity listed on the left with its corresponding classification and physical description on the right.

Click a left item, then click its matching right item

Items

Electric potential difference
Thermodynamic temperature
Impulse
Luminous intensity

Matches

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Answer

Electric potential difference matches derived quantity defined as work done per unit electric charge; Thermodynamic temperature matches fundamental quantity representing thermal state, measured in kelvins; Impulse matches derived quantity defined as force times time interval; Luminous intensity matches fundamental quantity measuring perceived light power per unit solid angle.
Thermodynamic temperature and luminous intensity are two of the seven fundamental SI quantities. Electric potential difference and impulse are derived quantities because they are expressed through equations involving base physical quantities.

Step-by-Step Solution

1
Identify fundamental physical quantities
Thermodynamic temperature and luminous intensity are basic independent physical quantities defined by SI standards.
Fundamental quantities cannot be defined in terms of other physical quantities.
2
Identify derived physical quantities and their defining expressions
Electric potential difference (V=WQV = \frac{W}{Q}) and impulse (I=FΔtI = F \Delta t) are derived from basic quantities.
Derived quantities are defined by mathematical combinations of fundamental quantities.

Key Concept

Fundamental and Derived Quantities
Question 20Question →

Match each physical quantity listed on the left with its correct classification and SI unit definition on the right.

Click a left item, then click its matching right item

Items

Electric current
Electric potential difference
Luminous intensity
Specific heat capacity

Matches

Show answer & explanation

Answer

Electric current matches fundamental quantity measured in amperes (A); Electric potential difference matches derived quantity measured in volts (V) or kg⋅m2⋅s−3⋅A−1kg\cdot m^2\cdot s^{-3}\cdot A^{-1}; Luminous intensity matches fundamental quantity measured in candelas (cd); Specific heat capacity matches derived quantity measured in joules per kilogram per kelvin (J⋅kg−1⋅K−1J\cdot kg^{-1}\cdot K^{-1}).
Electric current and luminous intensity belong to the seven basic SI fundamental quantities with units ampere and candela respectively. Electric potential difference (V=W/QV = W/Q) and specific heat capacity (c=Q/(mΔT)c = Q / (m\Delta T)) are derived physical quantities expressed in terms of fundamental SI units.

Step-by-Step Solution

1
Identify the fundamental physical quantities from the list.
Electric current and luminous intensity are base SI quantities.
Fundamental quantities are independent basic quantities defined by international standards.
2
Identify the derived physical quantities and determine their SI units.
Electric potential difference and specific heat capacity are derived quantities.
Derived quantities are formed from combinations of fundamental quantities through mathematical relationships.
3
Pair each physical quantity on the left with its exact classification and unit definition on the right.
Match left_1 to right_1, left_2 to right_4, left_3 to right_3, and left_4 to right_2.
Direct comparison with standard SI fundamental and derived quantity definitions confirms these pairings.

Key Concept

Fundamental and Derived Quantities