Animated Solution for Physics - Physics and Measurement: Some physical quantities are given in Column I and some possible SI units in which these quantities may be expressed are given in Column II. Match the physical quantities in Column I with the units in Column II.

The journey to solving this matrix match question is like being a detective decoding a secret language. We are given a set of complex physical expressions in Column I and a set of seemingly random SI units in Column II. Our mission? To find the perfect match.

But how do we bridge the gap between abstract formulas and raw units? The answer lies in the universal translator of physics: Dimensional Analysis. By breaking down every single term into its fundamental dimensions—Mass ($M$), Length ($L$), and Time ($T$)—we can easily see which quantities are truly identical.

The Hidden Formulas

Let's start by analyzing the expressions in Column I. If you look closely, you'll realize that these aren't just random variables thrown together; they are pieces of very famous physics equations!

Analyzing (A): $G{M}_e{M}_s$ Does this look familiar? Imagine Newton's Law of Universal Gravitation: If we rearrange this equation to isolate our target expression, we get: Now, finding the dimensions is a breeze. We know that force ($F$) has dimensions $[{\text{MLT}}^{-2}]$ and distance squared ($r^2$) has dimensions $[{\text{L}}^2]$. Multiplying them together gives:

Analyzing (B): $\frac{3RT}{M}$ Recall the kinetic theory of gases. The root-mean-square (RMS) speed of gas molecules is given by: Squaring both sides reveals our expression: Since this is simply velocity squared, its dimensions are:

Analyzing (C): $\frac{F^2}{q^2{B}^2}$ This one screams electromagnetism! The magnetic Lorentz force on a moving charge is: Rearranging for velocity gives $v=\frac{F}{qB}$. If we square this entire equation, we get exactly the expression in the question: Once again, this is just velocity squared! The dimensions are:

Analyzing (D): $\frac{G{M}_e}{R_e}$ This term appears in gravitation, specifically in the formula for the orbital velocity of a satellite near the Earth's surface: Squaring it gives $v_o^2$. Just like the previous two, this is also velocity squared, yielding the dimensions:

The Raw Units

Now that we have the dimensional fingerprints of Column I, let's translate the units in Column II into dimensions.

Decoding (p): (Volt) (Coulomb) (Metre) In electrostatics, the work done in moving a charge is $W=qV$. Therefore, the product of Volts and Coulombs is simply Joules (the unit of energy). Now we have Joules $\times$ Metres. Energy has dimensions $[{\text{ML}}^2{\text{T}}^{-2}]$, so multiplying by another length $[\text{L}]$ gives:

Decoding (q): (Kilogram) (Metre)^3 (Second)^{-2} This one is a direct translation. Kilogram is Mass $[\text{M}]$, Metre${}^3$ is Length cubed $[{\text{L}}^3]$, and Second${}^{-2}$ is Time to the power of negative two $[{\text{T}}^{-2}]$. Combining them gives:

Decoding (r): (Metre)^2 (Second)^{-2} Another direct translation. Metre squared is $[{\text{L}}^2]$ and Second${}^{-2}$ is $[{\text{T}}^{-2}]$. This gives:

Decoding (s): (Farad) (Volt)^2 (Kilogram)^{-1} Recall the formula for the energy stored in a capacitor: $U=\frac{1}{2}C{V}^2$. This tells us that Capacitance (Farads) multiplied by Voltage squared (Volts${}^2$) results in Energy (Joules). So, we have Joules per Kilogram. Dividing the dimensions of energy $[{\text{ML}}^2{\text{T}}^{-2}]$ by mass $[\text{M}]$ leaves us with:

Matching Them Up

We've successfully translated everything! Let's look at the results: - (A) has dimensions $[{\text{ML}}^3{\text{T}}^{-2}]$, which perfectly matches (p) and (q). - (B), (C), and (D) all have dimensions $[{\text{L}}^2{\text{T}}^{-2}]$, which perfectly match (r) and (s).

By recognizing the hidden physics formulas and using dimensional analysis as our bridge, we turned a complex matching problem into a beautifully logical puzzle!