53. The position of a particle along the y-axis is y —— P t4 + Q. For the equation to be dimensionally consistent, the dimension of P in terms of length [L] and time [T] is

(A) LT^-1

(B) LT^-2

(C) LT^-3

(D) LT^-4

Dimensions of P in the Equation y = Pt⁴ + Q – Complete Dimensional Analysis with Detailed Explanation

Correct Answer

(D) LT⁻⁴

Step-by-Step Solution Using Dimensional Analysis

The left-hand side of the equation represents the position of the particle. Position is a measure of distance, so its dimension is simply:

[y] = [L]

According to the principle of dimensional homogeneity, every term on both sides of a physical equation must possess the same dimensions. Therefore, the term Pt⁴ must also have the dimension of length.

Time has the dimension:

[t] = [T]

Therefore,

[Pt⁴] = [P][T⁴]

Since this must equal the dimension of position,

[P][T⁴] = [L]

Rearranging the equation gives:

[P] = [L][T]⁻⁴

Hence, the dimension of P is:

[P] = LT⁻⁴

Therefore, the correct answer is Option (D).

Why Dimensional Consistency is Important

The principle of dimensional consistency states that only quantities having identical dimensions can be added or subtracted. This rule is one of the most fundamental checks used in Physics to verify equations. If even one term has different dimensions, the equation cannot represent a physically valid relationship.

In this question, both Pt⁴ and Q must have the same dimensions as y, which is length. This immediately allows us to calculate the dimension of the unknown constant P without requiring any additional information.

Detailed Explanation of Every Option

Option (A): LT⁻¹

This dimension corresponds to length divided by time, which represents the dimension of velocity. If P had this dimension, then multiplying it by t⁴ would produce LT³, which is not the dimension of position. Therefore, this option is incorrect.

Option (B): LT⁻²

This represents the dimension of acceleration multiplied by time. Multiplying LT⁻² by T⁴ gives LT² instead of L. Since the resulting dimension is different from position, this option is also incorrect.

Option (C): LT⁻³

Multiplying LT⁻³ by T⁴ results in LT, which again does not represent the dimension of length alone. Hence, this option violates dimensional consistency and is incorrect.

Option (D): LT⁻⁴

Multiplying LT⁻⁴ by T⁴ gives L, which perfectly matches the dimension of position. Therefore, this is the only dimensionally correct answer and satisfies the principle of dimensional homogeneity.

Key Concept Behind This Question

Whenever constants appear in equations involving powers of time, distance, velocity, or acceleration, their dimensions are determined by ensuring that every term in the equation has identical dimensions. This technique is known as dimensional analysis and is widely used in Physics to derive formulas, verify equations, and solve numerical problems quickly.

For equations involving addition or subtraction, always remember that every term must possess exactly the same dimensions. This simple rule allows many examination questions to be solved within a few seconds.

Final Answer

Dimension of P = LT⁻⁴

Correct Option: (D)