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

In an electrical circuit containing a resistor and a capacitor, the product $RC$ defines the time constant, which characterizes the speed at which the capacitor charges or discharges. Because resistance is the ratio of potential difference to current and capacitance is the ratio of charge to potential difference, multiplying these two quantities cancels out the voltage terms. This reduction leaves the relationship in terms of charge divided by current, which is equivalent to the unit of time, confirming that the dimensions of $RC$ are simply time.

The propagation of uncertainty in a calculated result depends on the powers to which its constituent variables are raised. Because kinetic energy is determined by the equation $K = \frac{1}{2}mv^2$, the relative error in energy is the sum of the relative error in mass and twice the relative error in speed, accounting for the square on the velocity term.

Applying the given values, we add the $2\%$ error from the mass measurement to twice the $3\%$ error associated with the speed. This arithmetic leads to $2\% + 2 \times 3\%$, giving an overall uncertainty of $8\%$ for the kinetic energy estimate.

Dimensional analysis relies on the principle of homogeneity, which dictates that quantities can only be subtracted if they share the same dimensions, ensuring the term $(r^{2}-x^{2})$ possesses the dimensions $[L^{2}]$. To determine the dimensions of $\eta$, we substitute the fundamental dimensions of each variable into the provided formula $\eta = \frac{P(r^{2}-x^{2})}{4 v l}$, while noting that the numerical constant $4$ is dimensionless.

Substituting the known dimensions where $P$ is $[M L^{-1} T^{-2}]$, $r$ and $x$ are $[L]$, $v$ is $[L T^{-1}]$, and $l$ is $[L]$, the expression becomes \eta = \frac{[M L^{-1} T^{-2}]} $[L^{2}]}{[L T^{-1}]\{ [L]}$. Simplifying the numerator results in$[M L^{1} T^{-2}]$, while the denominator simplifies to$[L^{2} T^{-1}]$. Dividing the numerator by the denominator cancels out the length terms to produce$[M L^{1-2} T^{-2+1}]$, which yields the final dimensions of$[M L^{-1} T^{-1}]$.

The fundamental principle of dimensional analysis involves breaking down complex physical quantities into their base dimensions of mass, length, and time to identify unique properties. When we calculate the dimensions for energy per unit volume, force per unit area, and the product of voltage and charge divided by volume, we find that each simplifies to the expression $[ML^{-1}T^{-2}]$, which represents the dimensional formula for pressure. In contrast, angular momentum results in dimensions of $[ML^2T^{-1}]$. Since this specific result differs from the consistent dimensional formula shared by the other three quantities, it is distinct from the group.

To find the dimensional formula for the permeability of free space $\mu_{0}$, we start with the defining relationship derived from the magnetic force between wires, rearranged as $\mu_{0} = \frac{2\pi \cdot \text{force} \cdot \text{distance}}{\text{current} \cdot \text{current} \cdot \text{length}}$. Substituting the known dimensions for these quantities-force as $[MLT^{-2}]$, distance and length as $[L]$, and current as $[A]$-leads to the expression \frac{[MLT^{-2}]}[L]}{[A][A][L]}$. Simplifying this by canceling the redundant length terms gives us the final dimensions of$[MLT^{-2}A^{-2}]$.

Dimensional analysis relies on the principle of homogeneity, which states that for a physical expression to be dimensionless, the net exponent of each fundamental dimension-mass ($M$), length ($L$), and time ($T$)-must equal zero. First, express the given quantities in terms of fundamental units: radiation pressure $P$, which is force per unit area, corresponds to $[M L^{-1} T^{-2}]$, radiation intensity $S$, defined as energy per unit area per unit time, corresponds to $[M T^{-3}]$, and the speed of light $c$ corresponds to $[L T^{-1}]$. Equating the total dimensions of the expression $P^{x} S^{y} c^{z}$ to $[M^{0} L^{0} T^{0}]$ yields $[M L^{-1} T^{-2}]^{x}$[M T^{-3}]^{y} $[L T^{-1}]^{z} = [M^{0} L^{0} T^{0}]$. By equating the powers of the fundamental dimensions on both sides, we generate a system of simultaneous linear equations:$x + y = 0$for mass,$-x + z = 0$for length, and$-2x - 3y - z = 0$for time. Solving these equations shows that$y = -x$and$z = x$, leading to the values$x = 1$,$y = -1$, and$z = 1$.

Dimensional analysis relies on the principle that the dimensions on both sides of a valid physical equation must be identical. Since frequency corresponds to the inverse of time, its dimensions are $[M^0 L^0 T^{-1}]$, while the mass $m$ is simply $[M]$, and the spring constant $k$, representing force per unit length, is dimensionally equivalent to $[MT^{-2}]$.

Substituting these dimensions into the expression $f = a m^x k^y$ results in the equation $[M^0 L^0 T^{-1}] = [M]^x$[MT^{-2}]^y$. By combining the powers of the mass and time bases on the right side, the expression simplifies to$[M^{x+y} T^{-2y}]$. Matching the exponents of the corresponding bases on both sides yields the system of equations$x + y = 0$and$-2y = -1$. Solving these simultaneously gives$y = \frac{1}{2}$, which then determines that$x = -\frac{1}{2}$.

The voltage across a resistor is given by $V = iR$, while the voltage across an inductor is expressed as $V = L \frac{di}{dt}$. Equating these dimensions gives $[i][R] = [L] \frac{[i]}{[t]}$, where $i$ represents current and $t$ represents time. Canceling the current dimension $[i]$ from both sides leaves $[R] = \frac{[L]}{[t]}$, demonstrating that the ratio $\frac{L}{R}$ corresponds directly to the dimension of time $T$.

Determining the dimension of a complex physical expression involves multiplying the dimensional formulas of the individual constants and simplifying the resulting powers. Using the standard dimensions for the Stefan-Boltzmann constant $[M T^{-3} K^{-4}]$, the Boltzmann constant $[M L^2 T^{-2} K^{-1}]$, and Wien's displacement constant $[L K]$, we substitute these into the expression $\sigma_s k_B^{-1} b$. This results in $[M T^{-3} K^{-4}] \times$[M^{-1} L^{-2} T^2 K]$ \times [L K]$, where the Boltzmann constant is inverted due to its negative exponent. Combining the terms for each base dimension by summing their respective powers leads to the cancellation of mass, yielding the final dimensional formula$[L^{-1} T^{-1} K^{-2}]$.

Physical constants are categorized based on whether they possess dimensions or are merely unitless ratios. Quantities such as relative density, refractive index, and Poisson's ratio are all defined as ratios of similar physical properties, causing their units to cancel out entirely and leaving them dimensionless. In contrast, the gravitational constant is a dimensional constant because it retains specific units alongside its fixed numerical value to satisfy the physical requirements of equations like $F = G \frac{m_1 m_2}{r^2}$.

The least count of a Vernier calliper represents the smallest measurement it can resolve, calculated by dividing the main scale division by the total number of Vernier scale divisions, which gives $\frac{1 \text{ mm}}{10} = 0.1 \text{ mm} = 0.01 \text{ cm}$. When the Vernier scale zero is shifted to the left of the main scale zero, a negative zero error exists, quantified as $-(N - n) \times \text{LC}$, where $N$ is the total divisions and $n$ is the coinciding division. Applying the values, the zero error is $-(10 - 4) \times 0.01 \text{ cm} = -0.06 \text{ cm}$. Since the zero correction is defined as the negative of the zero error, the correction applied to the measurement is $+0.06 \text{ cm}$. Adding this correction to the observed measurement of $1 \text{ cm}$ yields an actual length of $1 + 0.06 = 1.06 \text{ cm}$.

The relationship between distance, speed, and time is defined by the formula $d = v \times t$. Since the speed of light is taken as 1 unit per second, the distance is numerically equal to the travel time in seconds. Converting 6 minutes and 40 seconds involves multiplying 6 by 60 to get 360 seconds and adding the remaining 40 seconds, which results in a total of 400 seconds. Using the values $v = 1$ and $t = 400$ in the equation $d = v \times t$, we obtain $d = 1 \times 400 = 400$ new units.

Magnetic flux represents the product of magnetic field strength and the surface area it penetrates, which is expressed by the relationship $\phi = B A$. Since the magnetic field is derived from the Lorentz force formula $F = IlB$, we determine its dimensions as $B = \frac{F}{Il}$, which simplifies to $[MT^{-2}A^{-1}]$ by using the fundamental dimensions for force, current, and length. Multiplying this value by the dimensions of area, $[L^2]$, results in the final dimensional formula for magnetic flux, $[ML^2T^{-2}A^{-1}]$.

Pressure is defined as the magnitude of force exerted perpendicular to a surface divided by the area over which that force is spread, expressed by the relationship $P = \frac{F}{A}$. By substituting the standard dimensional formulas for force, $[\text{MLT}^{-2}]$, and area, $[\text{L}^2]$, we obtain the ratio $[\text{MLT}^{-2}] / [\text{L}^2]$. Simplifying the exponents for length according to the laws of algebra yields the final dimensions of $[\text{ML}^{-1}\text{T}^{-2}]$.

The density of a material is defined by the formula $\rho = \frac{m}{V}$, where $V = a^3$ represents the volume of the cube with side length $a$. When calculating this, the significant figures in the result must match the measurement with the least precision, which is the side length of 9.0 cm, providing only two significant figures.

Convert the side length to meters to obtain $a = 9.0 \times 10^{-2}$ meters. Calculate the volume as $V = (9.0 \times 10^{-2})^3 = 7.29 \times 10^{-4}$ cubic meters. Dividing the mass of 5.580 kilograms by this volume gives $\rho = \frac{5.580}{7.29 \times 10^{-4}} = 7.654 \times 10^3$ kilograms per cubic meter. Rounding this result to two significant figures to respect the precision of the side length measurement yields 7.7, which is the value of $X$.

Self-inductance characterizes the proportionality between an induced electromotive force and the rate of change of current through the relation $\epsilon = L \frac{dI}{dt}$. Expressing this in terms of $L$ requires dividing the dimensions of electromotive force, $[ML^2T^{-3}A^{-1}]$, by the dimensions of the rate of change of current, $[AT^{-1}]$. This calculation produces the resulting dimensional formula $[ML^2T^{-2}A^{-2}]$.

In any exponential function, the argument serving as the exponent must be a dimensionless quantity. In the given expression $p = p_0 \exp(-\alpha t^{2})$, the term $\alpha t^{2}$ acts as this exponent, which necessitates that the product of $\alpha$ and $t^{2}$ has no physical dimensions. Since the variable $t$ represents time with dimensions of $[T]$, the constant $\alpha$ must inherently possess dimensions of $[T^{-2}]$ to ensure the entire exponent is dimensionless.

The physical quantity known as angular momentum can be calculated by multiplying the moment of inertia of a system by its angular velocity. In terms of dimensions, the moment of inertia represents mass distributed at a specific distance from an axis, yielding $[ML^2]$, whereas angular velocity represents an angle per unit time, which reduces to $[T^{-1}]$ because angles are dimensionless quantities. Combining these terms leads to the comprehensive dimensional formula for angular momentum, which is represented as:

[ML^2]$[T^{-1}]\$ = [ML^2T^{-1}]

The principle of dimensional homogeneity requires that every term in a physical equation must share the same dimensions for the expression to be consistent. In the equation $x = at + bt^2$, where $x$ represents distance in km and $t$ represents time in s, the term $bt^2$ must possess the same units as $x$ to be valid. By rearranging this relationship as $b = \frac{x}{t^2}$, we see that the unit of $b$ is derived by dividing the unit of distance by the square of the unit of time. Substituting the given units into the formula, we find that the unit of $b$ is km/s^2.

In electrical circuits, the product of resistance and capacitance functions as the time constant, indicating that it fundamentally represents a measure of time. By substituting the basic definitions of these components, where capacitance is charge divided by potential and resistance is potential divided by current, the product is calculated as:

$CR = \left( \frac{Q}{V} \right) \times \left( \frac{V}{I} \right) = \frac{Q}{I}$

Since electric current is defined as the rate of flow of charge, expressed as $I = Q/t$, substituting this into the previous equation results in $Q / (Q/t)$, which simplifies to $t$. This identifies the product as a time quantity, yielding the dimensional formula $[M^0 L^0 T^1]$.