Problem 40 - Master Resource Book in JEE Main Physics

Question

Match the physical quantities given in column I with dimension expressed in terms of mass \((m)\), length \((L)\), time ( \(T\) ) and change \((Q)\) given in column II. Column I \(\quad\) Column II (A) Angular momentum (p) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) (B) Torque (q) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-1}\right.\) ] (C) Inductance (r) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{-2} \mathrm{~T}^{2} \mathrm{Q}^{2}\right]\) (D) Latent heat (s) \(\left[\mathrm{ML}^{2} \mathrm{Q}^{-2}\right]\) (E) Capacitance (t) \(\left[\mathrm{ML}^{3} \mathrm{~T}^{-1} \mathrm{Q}^{-2}\right]\) (F) Resistivity (u) \(\left[\mathrm{L}^{2} \mathrm{~T}^{-2}\right]\) \(\begin{array}{llllll}\mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} & \mathrm{F}\end{array}\) (a) q s p t r u (b) q \(\quad \mathrm{p} \quad \mathrm{s} \quad \mathrm{u} \quad \mathrm{r} \quad \mathrm{t}\) (c) p \(\begin{array}{llllll}\text { (c) } & \text { u } & \text { r } & \text { t } & q\end{array}\) (d) \(\mathrm{s} \quad \mathrm{u}\) \(r\) \(\mathrm{t} \quad \mathrm{q} \quad \mathrm{p}\)

Step-by-Step Solution

Answer

(b) q, p, r, u, s, t.

Step 1: Understand the Problem

We're tasked with matching each physical quantity from Column I to its corresponding dimension in terms of mass \((m)\), length \((L)\), time \((T)\), and charge \((Q)\) listed in Column II. Each physical quantity has a unique dimension that we need to identify and match.

Step 2: Analyze the Physical Quantities

Each physical quantity represents a measurable aspect of physics and corresponds to a specific dimension. These quantities and their associated dimensions need to be understood.- Angular momentum is usually expressed as the product of mass, velocity, and distance from the point of rotation, leading to a dimension of \([\mathrm{ML}^2 \mathrm{T}^{-1}]\).- Torque, as a rotational force, is expressed similarly to energy or work but involves a rotational element, thus \([\mathrm{ML}^2 \mathrm{T}^{-2}]\).- Inductance, in an electrical context, is the ability of a coil to induce voltage when the electric current changes, measured in terms of \([\mathrm{M}^{-1} \mathrm{L}^{-2} \mathrm{T}^2 \mathrm{Q}^2]\).- Latent heat has dimensions that relate energy per unit mass, resulting in \([\mathrm{L}^2 \mathrm{T}^{-2}]\).- Capacitance, which is the ability to hold charge, involves dimensions expressed as \([\mathrm{M}^{-1} \mathrm{L}^{-2} \mathrm{T}^4 \mathrm{Q}^2]\).- Resistivity is derived from resistance per unit length, relating to \([\mathrm{ML}^{3} \mathrm{T}^{-1} \mathrm{Q}^{-2}]\).

Step 3: Match Physical Quantities to Dimensions

Using the analyzed dimensions:- A (Angular momentum) matches with (q) \([\mathrm{ML}^2 \mathrm{T}^{-1}]\).- B (Torque) matches with (p) \([\mathrm{ML}^2 \mathrm{T}^{-2}]\).- C (Inductance) matches with (r) \([\mathrm{M}^{-1} \mathrm{L}^{-2} \mathrm{T}^2 \mathrm{Q}^2] \).- D (Latent heat) matches with (u) \([\mathrm{L}^2 \mathrm{T}^{-2}]\).- E (Capacitance) matches with (s) \([\mathrm{M}^{-1} \mathrm{L}^{-2} \mathrm{T}^4 \mathrm{Q}^2]\).- F (Resistivity) matches with (t) \([\mathrm{ML}^{3} \mathrm{T}^{-1} \mathrm{Q}^{-2}]\).

Step 4: Select the Correct Option

Based on the matches found, the sequence of matches corresponds to option: (q - p - r - s - u - t), which is (b) in the provided options.

Key Concepts

Angular Momentum

Angular momentum represents the quantity of rotation of an object and is an essential concept in dynamics. The angular momentum of an object is given by the product of its moment of inertia and its angular velocity. In simpler terms, it can also be thought of as how much mass is rotating and how fast it is rotating. Mathematically, it is expressed as:

• Angular Momentum, \( L = I \cdot \omega \)
• where \( I \) is the moment of inertia, and \( \omega \) is the angular velocity.

The dimensions of angular momentum are \( \left[ ML^2 T^{-1} \right] \). This reflects its dependence on factors like mass, length, and time. It plays a crucial role in rotational systems, where conserving angular momentum can simplify a lot of complex situations.

Torque

Torque, often referred to as the rotational equivalent of force, describes how a force causes an object to rotate. The larger the torque applied, the greater the change in the rotational state of the object.
Mathematically, torque can be represented as:

• Torque, \( \tau = r \times F \)
• where \( r \) is the distance vector from the pivot point to the point where the force is applied, and \( F \) is the force vector.

The dimension of torque is \( \left[ ML^2 T^{-2} \right] \), indicating how it relates to mass, length, and time units associated with rotational dynamics. Torque is foundational in understanding rotational motion, especially in mechanics.

Inductance

Inductance is a property of an electrical circuit or component which induces electromotive force when the current flowing through it changes. It essentially opposes changes in the current. A common device using inductance is a coil or inductor.
The formula related to inductance is:

• EMF induced, \( \mathcal{E} = -L \frac{dI}{dt} \)
• where \( L \) is inductance, \( dI/dt \) is the rate of change of current.

Its dimension is represented as \( \left[ M^{-1} L^{-2} T^2 Q^2 \right] \), illustrating its dependence on mass, length, time, and charge. Inductance is crucial in circuits, especially those dealing with alternating current (AC).

Latent Heat

Latent heat is the amount of heat absorbed or released during a phase change of a substance, without changing its temperature. This concept is essential when studying changes of state, such as melting or boiling.

• Formula: \( Q = mL \)
• where \( Q \) is the heat energy, \( m \) is the mass, and \( L \) is the latent heat.

The dimensions of latent heat are \( \left[ L^2 T^{-2} \right] \), reflecting its relationship with length and time. Understanding latent heat helps in many applications, including thermal management and designing systems for heating or cooling.

Capacitance

Capacitance is the capability of a system to store an electric charge. This property is found in capacitors, which are widely used components in electrical circuits.

• Capacitance, \( C = \frac{Q}{V} \)
• where \( Q \) is the charge stored and \( V \) is the voltage across the capacitor.

Its dimension is given by \( \left[ M^{-1} L^{-2} T^4 Q^2 \right] \), linking it to mass, length, time, and charge. Capacitance is foundational for electronics as it impacts circuit behavior, volley of energy storage, and filtering applications.

Resistivity

Resistivity is a fundamental property that quantifies how strongly a material opposes the flow of electric current. It's an intrinsic property, meaning it doesn't depend on the shape or size of the material.

• The formula is \( \rho = R \frac{A}{l} \)
• where \( \rho \) is resistivity, \( R \) is resistance, \( A \) is cross-sectional area, and \( l \) is length.

Resistivity is dimensioned as \( \left[ ML^3 T^{-1} Q^{-2} \right] \), which relates to mass, length, time, and charge. It's crucial to understand resistivity for material selection in electronics and designing efficient electrical circuits.