Physical World Units And Measurements – NEET UG PhysicsPractice Questions & PYQs

The concepts of plane angle and solid angle are defined by ratios of identical physical dimensions, specifically represented as $\text{Plane angle} = \frac{[L]}{[L]}$ and \text{Solid angle} = \frac{[L^2]}}{[L^2]${}$. Since dividing like physical dimensions causes them to cancel out completely, both quantities result in a dimensionless value of$[M^0L^0T^0]$. Despite this lack of dimensions, these quantities remain physically significant and are assigned specific units, known as the radian for plane angle and the steradian for solid angle.

Significant figure rules for multiplication require the product to be limited by the measurement with the fewest significant figures. Since the length 55.3 m has three significant figures and the breadth 25 m has two, the final area must be rounded to two significant figures. The raw product is $55.3 \times 25 = 1382.5 \text{ m}^{2}$, which rounds to 1400 when restricted to two digits. This leads to an area of $14 \times 10^{2} \text{ m}^{2}$.

Dimensional analysis simplifies complex quantities by expressing them in terms of fundamental mass, length, and time units. Pressure is physically defined as force applied per unit area, which yields a dimensional formula of $[ML^{-1}T^{-2}]$. Since Young's modulus of elasticity is defined as the ratio of stress to strain and strain itself is a dimensionless quantity, the dimensions of Young's modulus simplify to the dimensions of stress, which is also force per unit area. Consequently, both physical quantities share the exact same dimensional representation of $[ML^{-1}T^{-2}]$.

When determining the error in a derived quantity, the relative uncertainties of the individual measurements are combined, with any exponents acting as multipliers for their respective fractional errors. For the formula $g = \frac{4 \pi^2 L}{T^2}$, the constant term $4 \pi^2$ is precise and does not contribute to the overall uncertainty. The total percentage error in $g$ is found by adding the percentage error of $L$ to twice the percentage error of $T$, as expressed in the equation below:

$\frac{\Delta g}{g} \times 100 = \left( \frac{\Delta L}{L} \times 100 \right) + 2 \left( \frac{\Delta T}{T} \times 100 \right)$

Using the given measurements where $L$ is 10 ± 0.1 cm and $T$ is 100 ± 1 s, the percentage error contribution from $L$ is 0.1 divided by 10, multiplied by 100, which equals 1%. The contribution from $T$ is calculated by multiplying 2 by the fractional error of 1 divided by 100, then multiplying by 100, resulting in 2%. Adding these two contributions together yields a total percentage error of 3%.

Dimensional analysis allows us to determine the units of a composite quantity by dividing the fundamental dimensions of its constituents. Energy is characterized by the dimensional formula $[M^1 L^2 T^{-2}]$, while the gravitational constant is defined by $[M^{-1} L^3 T^{-2}]$. Dividing these expressions requires subtracting the exponent of the denominator from the exponent of the numerator for each base dimension:

\frac{[M^1 L^2 T^{-2}]}$}{[M^{-1} L^3 T^{-2}]\${} = [M^{1 - (-1)} L^{2 - 3} T^{-2 - (-2)}]

This arithmetic yields the final dimensional formula of $[M^2 L^{-1} T^0]$.

Measurement with a screw gauge relies on determining its least count, which represents the smallest measurable distance on the tool. This is calculated by dividing the pitch by the total number of circular scale divisions, resulting in $LC = \frac{1 \text{ mm}}{100} = 0.01 \text{ mm}$. Converting this to centimeters gives 0.001 cm. The total diameter of the wire is obtained by adding the main scale reading to the product of the circular scale reading and the least count using the formula $D = \text{MSR} + (\text{CSR} \times LC)$. Substituting the values into this expression results \in $0 + (52 \times 0.001 \text{ cm}) = 0.052 \text{ cm}$.

Dimensional analysis relies on the principle that the physical dimensions on both sides of an equation must balance, so we assume $[E] = [F]^a [A]^b [T]^c$ and substitute the standard dimensions:
[M L^2 T^{-2}] = $[M L T^{-2}]^a \$[L T^{-2}]^b [T]^c
Simplifying the right side results in $[M^a L^{a+b} T^{-2a-2b+c}]$, which allows us to equate the powers to form the system $a = 1$, $a + b = 2$, and $-2a - 2b + c = -2$. Solving these equations gives $a = 1$, $b = 1$, and $c = 2$, leading to the final dimensional form of $[F A T^2]$.

The least count of a screw gauge, which represents the smallest measurement it can accurately record, is determined by dividing the pitch by the total number of divisions on the circular scale. This fundamental relationship is expressed by the formula $LC = \frac{\text{Pitch}}{\text{Number of divisions}}$. Given the least count of 0.01 mm and 50 circular scale divisions, multiplying these values confirms that the pitch of the screw gauge is 0.5 mm.

When performing subtraction, the precision of the result is limited by the measurement with the fewest decimal places. Calculating the raw difference gives $9.99 - 0.0099 = 9.9801$. Since the value 9.99 has only two decimal places, the final result must be rounded to two decimal places, giving 9.98 m.

Stress is defined as the internal restoring force acting over a cross-sectional area. Finding the dimensions requires dividing the force, $[MLT^{-2}]$, by the area, $[L^2]$. This division leads to the expression \frac{[MLT^{-2}]}}{[L^2]${}$, which simplifies to$[ML^{-1}T^{-2}]$ according to the laws of exponents.

The ability of a material to conduct heat is defined by the relationship $\frac{\mathrm{d}H}{\mathrm{d}t} = \frac{K A}{l} \Delta T$, where the rate of heat transfer is proportional to cross-sectional area and the temperature difference across the conductor. Rearranging this equation to isolate the coefficient of thermal conductivity yields $K = \frac{l (\mathrm{d}H/\mathrm{d}t)}{A \Delta T}$, where substituting the SI units of meters for length, watts for heat flow, square meters for area, and Kelvin for temperature difference is required. This calculation results in the unit of W m^-1 K^-1.

The least count of a vernier caliper is the difference between one main scale division and one vernier scale division. Given the main scale has $n$ divisions per centimeter, one main scale division is $\frac{1}{n}$ cm. Since $n$ vernier divisions coincide with $(n-1)$ main scale divisions, one vernier division equals $\frac{n-1}{n}$ main scale divisions, which makes the least count $\frac{1}{n}$ of a main scale division. Substituting the known value of the main scale division into this relationship provides the final result:
$\frac{1}{n} \times \frac{1}{n} \text{ cm} = \frac{1}{n^2} \text{ cm}$

When determining the maximum percentage error for a quantity defined by exponents, such as $X = \frac{A^2 B^{1/2}}{C^{1/3} D^3}$, one must calculate the sum of the individual percentage errors of each component, weighted by their respective power indices. This additive approach accounts for the cumulative effect of uncertainties across all variables, regardless of whether they appear in the numerator or denominator of the expression.

Applying this principle, the total error is the sum of the individual error contributions. The term $A^2$ contributes $2 \times 1\% = 2\%$. The term $B^{1/2}$ contributes $\frac{1}{2} \times 2\% = 1\%$. The term $C^{1/3}$ contributes $\frac{1}{3} \times 3\% = 1\%$, and the term $D^3$ contributes $3 \times 4\% = 12\%$. Summing these values, $2\% + 1\% + 1\% + 12\%$, results in a total maximum percentage error of $16\%$.

A screw gauge determines the diameter of an object by combining the reading from the main scale with the reading from the circular scale, then adjusting the result to account for the instrument's zero error. The main scale reading of 5 mm is equivalent to 0.5 cm. The contribution from the circular scale is calculated by multiplying the 25 divisions by the least count of 0.001 cm, which results in 0.025 cm.

Adding these two values provides an observed diameter of 0.525 cm. Since the instrument has a negative zero error of -0.004 cm, subtracting this error from the observed measurement yields the true diameter, as shown in the calculation: $0.5 + 0.025 - (-0.004) = 0.529$ cm.

Assigning fundamental dimensions to $c$, $G$, and $\frac{e^2}{4 \pi \varepsilon_0}$ reveals their values as $[LT^{-1}]$, $[M^{-1}L^3T^{-2}]$, and $[ML^3T^{-2}]$ respectively, where the latter is derived from the product of force and distance squared. Assuming the relationship follows the dimensional form $L = c^a G^b \left(\frac{e^2}{4 \pi \varepsilon_0}\right)^c$, we equate the powers of mass, length, and time across both sides. This generates the system of equations $-b + c = 0$, $a + 3b + 3c = 1$, and $-a - 2b - 2c = 0$. Solving this system yields $a = -2$, $b = 1/2$, and $c = 1/2$, providing the expression $\frac{1}{c^2}\left[G \frac{e^2}{4 \pi \varepsilon_0}\right]^{\frac{1}{2}}$.

Dimensional analysis relies on expressing physical constants in terms of fundamental units of mass ($M$), length ($L$), and time ($T$) to derive relationships between them. Planck's constant ($h$) has dimensions $[ML^2T^{-1}]$, the speed of light ($c$) is characterized by $[LT^{-1}]$, and the gravitational constant ($G$) is represented by $[M^{-1}L^3T^{-2}]$. Constructing a formula $h^x c^y G^z$ that yields the dimension of length $[L]$ involves solving for the exponents $x$, $y$, and $z$ by balancing these dimensions.

Equating the powers of mass, length, and time in the expression $[ML^2T^{-1}]^x$[LT^{-1}]^y $[M^{-1}L^3T^{-2}]^z = [L]$leads to a system of linear equations. For mass,$x - z = 0$, implying$x = z$. For time,$-x - y - 2z = 0$, which simplifies to$-z - y - 2z = 0$, giving$y = -3z$. Substituting these relationships into the equation for length,$2x + y + 3z = 1$, we get$2z - 3z + 3z = 1$, which simplifies to$2z = 1$, resulting in$z = 1/2$. Consequently,$x = 1/2$and$y = -3/2$. Substituting these values back into the general form yields$h^{1/2} c^{-3/2} G^{1/2}$, which is equivalent to$\frac{\sqrt{h G}}{c^{3/2}}$.

The principle of dimensional homogeneity serves as the foundation for this problem, requiring that the dimensions on both sides of the physical equation must be identical. We express the critical velocity as $[M^0 L T^{-1}]$ and equate it to the product of the given physical quantities raised to their unknown exponents, written as $[M L^{-1} T^{-1}]^x$[M L^{-3}]^y [L]^z$. By consolidating these terms, the right side becomes$[M^{x+y} L^{-x-3y+z} T^{-x}]$. Comparing the exponents for each base unit across both sides results \in a system of linear equations:$x + y = 0$,$-x - 3y + z = 1$, and$-x = -1$. Solving the third equation first gives$x = 1$, which leads directly to$y = -1$when substituted into the first equation, and finally yields$z = -1$ upon solving the second equation.

Newton's second law defines force as the product of mass and acceleration, providing the primary relationship needed to derive the dimensions of mass. Acceleration is characterized by the rate of change of velocity, given by the expression $A = \frac{V}{T}$ When this is substituted into the definition of force, the equation becomes $F = M \left( \frac{V}{T} \right)$ Isolating mass allows us to rearrange the terms to $M = F \cdot T \cdot V^{-1}$ confirming that the dimensions of mass are $[F V^{-1} T]$.

Physical quantities share identical dimensions when their expressions in terms of mass, length, and time reduce to the same base units. Work is defined as force multiplied by displacement, while torque is defined as force multiplied by a perpendicular distance, resulting in both quantities sharing the dimensional formula $[ML^2T^{-2}]$. Because the other listed pairings involve distinct physical definitions that lead to incompatible dimensions, work and torque are the only quantities with equivalent dimensional properties.

When a physical quantity is defined by a power-law relationship involving measured variables, the maximum relative error in the quantity is calculated by summing the products of the absolute values of the exponents and the relative errors of the individual measurements. For the specific expression $P=\frac{a^3b^2}{cd}$, the percentage error is given by the relation $\frac{\Delta P}{P} \times 100 = 3\left(\frac{\Delta a}{a}\times 100\right) + 2\left(\frac{\Delta b}{b}\times 100\right) + \left(\frac{\Delta c}{c}\times 100\right) + \left(\frac{\Delta d}{d}\times 100\right)$.

Substituting the provided percentage errors for $a$, $b$, $c$, and $d$ as $1\%$, $2\%$, $3\%$, and $4\%$ respectively, the equation becomes $3(1\%) + 2(2\%) + 3\% + 4\%$. This simplifies to $3\% + 4\% + 3\% + 4\%$, resulting in a total percentage error of $14\%$.