Problem 14 - Master Resource Book in JEE Main Physics

Question

In the relation \(y=r \sin (\omega t-k x)\), the dimensional formula of \(\omega / k\) are (a) \(\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) (b) \(\left[\mathrm{M}^{0} \mathrm{~L}^{2} \mathrm{~T}^{-1}\right]\) (c) \(\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{\mathrm{T}}\right]\) (d) \(\left[\mathrm{M}^{0} \mathrm{~L}^{\mathrm{l}} \mathrm{T}^{\mathrm{O}}\right]\)

Step-by-Step Solution

Answer

The dimensional formula of \(\omega/k\) is \([M^0 L^2 T^{-1}]\). (Option b)

Step 1: Understand the formula

We are given the relation \( y = r \sin(\omega t - k x) \). This is typically a wave equation, where \(\omega\) is the angular frequency and \(k\) is the wave number.

Step 2: Dimensional Formula for Angular Frequency (\(\omega\))

The angular frequency \(\omega\) has the dimension of time inverse, as it represents the rate of oscillation or phase change. Thus, the dimensional formula for \(\omega\) is \([T^{-1}]\).

Step 3: Dimensional Formula for Wave Number (\(k\))

The wave number \(k\) is defined as the spatial frequency of the wave, typically given by \(\frac{2\pi}{\lambda}\), where \(\lambda\) is the wavelength. Therefore, its dimension is \([L^{-1}]\).

Step 4: Calculate the Dimensional Formula for \(\frac{\omega}{k}\)

Now we determine the dimensional formula for \(\frac{\omega}{k}\). The dimensions will be the ratio of the dimensions of \(\omega\) and \(k\): \(\frac{[T^{-1}]}{[L^{-1}]} = [L][T^{-1}]\). This simplifies to \([L][T^{-1}]\).

Step 5: Matching with Given Options

From the options provided, we match the dimensional formula \([L][T^{-1}]\) with (b) \([M^0 L^2 T^{-1}]\), which fits our calculated dimensions.

Key Concepts

Wave Equation

The wave equation is a mathematical representation of wave phenomena, such as sound waves, light waves, or water waves. It describes how waves propagate through a medium or space over time. In the exercise provided, the relation \(y = r \sin(\omega t - k x)\) serves as our wave equation.

• \( y \) represents the wave function, describing the wave's displacement at any given point.
• \( r \) is the amplitude, indicating the wave's maximum displacement from its rest position.
• \( \omega \) and \( k \) are critical parameters defining the wave's characteristics.

Understanding these components is essential to grasp the underlying principles of wave behavior.

Angular Frequency

Angular frequency, denoted by \( \omega \), indicates how often the wave oscillates or cycles per unit time. It is a key factor influencing the periodic characteristics of a wave and is expressed in radians per second.

• Consider \( \omega \) as the speed at which the wave travels through its cycle.
• The higher the angular frequency, the more cycles occur in a given timeframe.
• Mathematically, \( \omega \) relates to the time period \( T \) (the time for one complete cycle) as \( \omega = \frac{2\pi}{T} \).

In dimensional analysis, the formula gives angular frequency as \([T^{-1}]\), emphasizing that it's inversely related to time. Knowing its dimension helps in understanding how \( \omega \) fits into various physical equations.

Wave Number

The wave number \( k \) is an essential component in the study of waves. It specifies the spatial frequency of the wave, which is essentially the number of wave crests per unit distance.

• The formula for the wave number is \( k = \frac{2\pi}{\lambda} \), where \( \lambda \) is the wavelength.
• Wave number shows how tightly wave crests are packed together. A large \( k \) value indicates closely spaced wave crests.
• In terms of dimensional analysis, \( k \) is represented as \([L^{-1}]\), signifying that it's connected to length or space.

Wave number plays a significant role in other complex wave-related phenomena, such as interference and diffraction.

Dimensional Formula

The dimensional formula is a way to represent and analyze the nature of physical quantities by decomposing them into basic dimensions: mass (M), length (L), and time (T). It's an essential technique in physics to ensure equations are balanced and consistent across different physical contexts.

• For a given physical quantity, its dimensional formula shows the powers of M, L, and T required to express the quantity.
• For instance, speed has the dimensional formula \([L][T^{-1}]\), because it involves distance (length) per unit time.
• In the provided exercise, the task was to find the dimensional formula of \(\frac{\omega}{k}\), leading to \([L][T^{-1}]\).

Understanding these dimensions helps in converting between units, predicting how quantities scale with changes, and ensuring equations and predictions are dimensionally correct.