Measurement/Unit Archives

1. Dioptre is the SI unit of :

Dioptre is the SI unit of :

size of the lens.

curvature of the lens.

power of the lens.

magnification produced by a lens.

This question was previously asked in

UPSC CISF-AC-EXE – 2022

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Dioptre (symbol D) is the unit of measurement for the optical power of a lens or curved mirror.
Optical power (P) of a lens is defined as the reciprocal of its focal length (f) when the focal length is measured in metres.
P = 1 / f (where f is in metres)
If the focal length f is in metres, the unit of power P is m⁻¹, which is defined as the dioptre (D).
For example, a lens with a focal length of +0.5 metres has a power of P = 1 / 0.5 m = +2 D.
A lens with a focal length of -0.25 metres has a power of P = 1 / -0.25 m = -4 D.

– Dioptre is the unit used to measure the optical power of lenses.
– Optical power is the reciprocal of the focal length measured in metres.

Size of the lens is measured in units of length (e.g., diameter in cm). Curvature of the lens is related to the radii of curvature of its surfaces, usually measured in reciprocal metres (m⁻¹), and contributes to the power of the lens along with the refractive index of the material. Magnification is a ratio of image size to object size, which is a dimensionless quantity or expressed as ‘x’. Dioptre specifically quantifies the power of the lens to converge or diverge light.

2. Which one of the following pairs of quantities has no length in their

Which one of the following pairs of quantities has no length in their dimension?

Surface tension and angular momentum

Surface tension and strain

Angular momentum and mass density

Pressure gradient and angle

This question was previously asked in

UPSC CISF-AC-EXE – 2019

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The correct answer is Surface tension and strain.

We need to find the pair of quantities whose dimensions do not include length ([L]). Let’s analyze the dimensions of each quantity:
– **Surface tension:** Defined as force per unit length. Dimension is [Force]/[Length] = [MLT⁻²]/[L] = [MT⁻²]. It has no length dimension.
– **Angular momentum:** Defined as the product of moment of inertia and angular velocity (Iω) or cross product of position and linear momentum (r x p). Dimension is [ML²] * [T⁻¹] = [ML²T⁻¹]. It has a length dimension.
– **Strain:** Defined as the ratio of change in dimension to the original dimension (e.g., change in length / original length). Dimension is [L]/[L] = [Dimensionless]. It has no length dimension.
– **Mass density:** Defined as mass per unit volume. Dimension is [M]/[L³] = [ML⁻³]. It has a length dimension.
– **Pressure gradient:** Defined as change in pressure per unit distance. Dimension is [Pressure]/[Length]. Pressure is [Force]/[Area] = [MLT⁻²]/[L²] = [ML⁻²T⁻²]. So, pressure gradient is [ML⁻²T⁻²]/[L] = [ML⁻³T⁻²]. It has a length dimension.
– **Angle:** Defined as the ratio of arc length to radius. Dimension is [L]/[L] = [Dimensionless]. It has no length dimension.

Now let’s check the pairs:
A) Surface tension ([MT⁻²]) and angular momentum ([ML²T⁻¹]). Angular momentum has length.
B) Surface tension ([MT⁻²]) and strain ([Dimensionless]). Neither has length.
C) Angular momentum ([ML²T⁻¹]) and mass density ([ML⁻³]). Both have length.
D) Pressure gradient ([ML⁻³T⁻²]) and angle ([Dimensionless]). Pressure gradient has length.

The pair with no length dimension is Surface tension and Strain.

Understanding the dimensional formulas of common physical quantities is crucial for solving such problems. Dimensional analysis helps verify the consistency of equations and understand the fundamental nature of physical quantities.

3. Which of the following statements regarding temperature of an object i

Which of the following statements regarding temperature of an object in Kelvin scale is/are correct ?

1. It can be a negative, zero or positive quantity.
2. It can either be a negative or a positive quantity.
3. It can never be negative.
4. It can be a positive definite quantity.

Select the correct answer using the code given below :

1 and 2 only

2 and 4 only

3 and 4 only

4 only

This question was previously asked in

UPSC CISF-AC-EXE – 2018

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The Kelvin scale is an absolute temperature scale. The zero point (0 K) is absolute zero, the theoretical lowest possible temperature. Temperatures on the Kelvin scale are always non-negative (greater than or equal to zero).
Statement 1: Incorrect, as Kelvin temperature cannot be negative.
Statement 2: Incorrect, as Kelvin temperature cannot be negative and can be zero.
Statement 3: Correct, Kelvin temperature can never be negative (it is always $\ge 0$).
Statement 4: Interpreting “positive definite quantity” in this context to mean a quantity that is positive or zero (non-negative), this statement is correct, as Kelvin temperature is always $\ge 0$. If interpreted strictly as strictly positive (> 0), it would be incorrect as 0 K is possible. Given the options, the non-negative interpretation is most likely intended for statement 4, making both 3 and 4 correct.

Temperature on the Kelvin scale is always non-negative ($\ge 0$).

Absolute zero (0 K or -273.15 °C) is the point at which particles have minimum possible motion (though not zero motion according to quantum mechanics). The Kelvin scale is used extensively in scientific applications.

4. Which one of the following is a reason why astronomical distances are

Which one of the following is a reason why astronomical distances are measured in light-years?

Distances among stellar bodies do not change.

Gravity of stellar bodies does not change.

Light always travels in straight line.

Speed of light is always same.

This question was previously asked in

UPSC IAS – 2021

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The correct answer is D) Speed of light is always same.

Astronomical distances are vast, making standard units like kilometres or miles impractical. A light-year is defined as the distance light travels in one year in a vacuum. The use of light-years as a unit of distance in astronomy is based on the fact that the speed of light in a vacuum is a constant (approximately 299,792,458 meters per second). This constant speed provides a reliable basis for measuring extremely large distances across space based on time.

Options A, B, and C are incorrect. Distances between celestial bodies do change (due to cosmic expansion, orbital motion, etc.). Gravity of stellar bodies also changes depending on mass distribution and distance. While light travels in approximately straight lines in uniform media, it can be bent by gravity (gravitational lensing). The constancy of the speed of light is the fundamental principle behind using light-years to measure distance.

5. Which one of the following is not the unit of pressure?

Which one of the following is *not* the unit of pressure?

pascal (Pa)

N/m²

J/m²

bar

This question was previously asked in

UPSC CAPF – 2022

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The correct answer is C, J/m².

Pressure is defined as force per unit area (Pressure = Force/Area).
The SI unit of force is Newton (N) and the SI unit of area is square meter (m²). Therefore, N/m² is a unit of pressure.
Pascal (Pa) is the SI unit of pressure, defined as 1 Pa = 1 N/m².
Bar is a non-SI unit of pressure, widely used (1 bar = 10⁵ Pa).
Joule (J) is the unit of energy or work. J/m² represents energy per unit area. While pressure has the same dimensions as energy density (J/m³), J/m² represents pressure multiplied by distance (since J = N.m, J/m² = (N.m)/m² = N/m * m = Pressure * Distance). Therefore, J/m² is not a unit of pressure.

J/m³ (Joule per cubic meter) represents energy density and has the same units as pressure (N/m² or Pa). This is because Energy = Force × Distance, so J = N.m. Thus, J/m³ = N.m/m³ = N/m², which are units of pressure. However, J/m² is not dimensionally equivalent to pressure.

6. Consider the following physical quantities : Energy, power, pressure,

Consider the following physical quantities :
Energy, power, pressure, impulse, temperature, gravitational potential
Which of the above is / are the vector quantity/quantities ?

Impulse only

Impulse and pressure only

Impulse, temperature and pressure

Gravitational potential

This question was previously asked in

UPSC CAPF – 2016

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Among the given physical quantities, only Impulse is a vector quantity. A vector quantity has both magnitude and direction. Impulse is defined as the change in momentum of an object, and momentum (mass × velocity) is a vector quantity. Impulse is also equal to the force multiplied by the time interval over which the force acts (when force is constant), and force is a vector.

Identifying whether a physical quantity is a vector or a scalar is a basic concept in physics.

Energy, power, pressure, temperature, and gravitational potential are all scalar quantities. Scalar quantities are completely described by their magnitude alone. While pressure involves force (a vector) acting over an area (direction can be associated with area normal), pressure itself at a point is a scalar property of the fluid or material. Gravitational potential is potential energy per unit mass, and potential energy is a scalar.

7. The radius of a hydrogen atom is 10 -10 m. Number of hydrogen atoms n

The radius of a hydrogen atom is 10^-10 m. Number of hydrogen atoms necessary to have a length of one nanometre is :

6.023 × 10<sup>23</sup>

100

This question was previously asked in

UPSC CAPF – 2015

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The radius of a hydrogen atom is given as 10⁻¹⁰ m. We want to find how many hydrogen atoms are needed to form a length of one nanometre (1 nm). One nanometre is equal to 10⁻⁹ metres. To arrange atoms side-by-side to achieve a certain length, we consider their effective diameter. The diameter of a hydrogen atom is approximately 2 times its radius, i.e., 2 * 10⁻¹⁰ m. The number of atoms needed is the total length divided by the diameter of one atom: Number of atoms = (10⁻⁹ m) / (2 * 10⁻¹⁰ m) = (10⁻⁹ / 10⁻¹⁰) / 2 = 10 / 2 = 5.

When stacking spherical objects (like atoms) linearly, the effective length occupied by each object is its diameter.

The size of atoms is typically on the order of angstroms (1 Å = 10⁻¹⁰ m). A nanometre (1 nm = 10⁻⁹ m = 10 Å) is a unit of length often used in nanoscience and technology. This problem provides a basic illustration of scale conversion between atomic dimensions and nanometre scale.

8. The amplitude of sound waves is measured in the units of

The amplitude of sound waves is measured in the units of

pressure

distance

time

speed

This question was previously asked in

UPSC NDA-2 – 2022

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The amplitude of sound waves is measured in the units of pressure.

Sound waves in a medium like air are longitudinal waves involving compressions and rarefactions. The amplitude of a sound wave is the maximum variation in pressure from the equilibrium pressure (atmospheric pressure) or the maximum displacement of particles from their mean position.

While amplitude can be expressed as displacement amplitude (measured in meters, a unit of distance), it is more commonly and directly related to the intensity or loudness of the sound when expressed as pressure amplitude. Pressure is measured in Pascals (Pa), which are units of pressure. The options provided include both pressure and distance, but pressure is a more direct measure of the force per unit area variation that constitutes the wave, and its amplitude correlates directly with loudness.

9. The unit of the ratio between thrust and impulse is same as that of

The unit of the ratio between thrust and impulse is same as that of

frequency

speed

wavelength

acceleration

This question was previously asked in

UPSC NDA-2 – 2021

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The correct option is A. We need to find the unit of the ratio between thrust and impulse and compare it with the units of the given options.

Thrust is a force. The SI unit of force is Newton (N), which is equivalent to $\text{kg} \cdot \text{m/s}^2$.
Impulse is the change in momentum, or force multiplied by time. The SI unit of impulse is N·s, which is equivalent to $(\text{kg} \cdot \text{m/s}^2) \cdot \text{s} = \text{kg} \cdot \text{m/s}$.
The ratio $\frac{\text{Thrust}}{\text{Impulse}}$ has units $\frac{\text{N}}{\text{N} \cdot \text{s}} = \frac{1}{\text{s}}$.
The unit $1/\text{s}$ is the unit of frequency. Frequency is the number of cycles or events per unit time, measured in Hertz (Hz), where 1 Hz = $1/\text{s}$.
Let’s check the units of the options:
A) Frequency: $1/\text{s}$ or Hz.
B) Speed: m/s.
C) Wavelength: m.
D) Acceleration: m/s².
The unit of the ratio is the same as the unit of frequency.

Impulse is also equal to the change in momentum ($\Delta p = m \Delta v$). So the unit of impulse is also $(\text{kg}) \cdot (\text{m/s}) = \text{kg} \cdot \text{m/s}$. Using this, the ratio unit is $\frac{\text{kg} \cdot \text{m/s}^2}{\text{kg} \cdot \text{m/s}} = \frac{1}{\text{s}}$.

10. Which one of the following is NOT the unit of energy ?

Which one of the following is NOT the unit of energy ?

Joule

Watt-hr

Newton-metre

kg-metre/sec²

This question was previously asked in

UPSC NDA-2 – 2020

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Let’s analyze the given units:
A) Joule (J) is the standard SI unit of energy and work.
B) Watt-hour (Wh) is a unit of energy, calculated as power (Watt) multiplied by time (hour). 1 Wh = 1 J/s * 3600 s = 3600 J.
C) Newton-metre (Nm) is the unit of work (Force x Distance). Work is a form of energy transfer, so Nm is equivalent to a Joule (1 Nm = 1 J). Note: Nm is also the unit of torque, but in the context of options alongside Joule and Watt-hour, it represents work/energy.
D) kg-metre/sec² is the unit of force (mass x acceleration, F = ma). 1 kg⋅m/s² = 1 Newton (N). It is not a unit of energy.

Energy can be expressed in various units depending on the context (e.g., Joules, kilowatt-hours, calories, foot-pounds). Force and Energy are distinct physical quantities with different units.

Energy can be defined as the capacity to do work. Work done by a force is the product of the force and the distance moved in the direction of the force (W = Fd). The unit of force is Newton (N), and the unit of distance is metre (m), so the unit of work (and energy) is N*m, which is defined as the Joule (J).