SSC Selection Post 2025 PYQ — Question 4

Correct Answer

Trigonometric and logarithmic functions are valid only when applied to quantities that are dimensionless

Explanation

According to the rules of dimensional analysis, the arguments or angles of trigonometric (such as sin, cos), logarithmic (log), and exponential functions must always be dimensionless. This is because these functions represent pure numbers or are based on the ratios of two identical quantities. For example, in sin(wt), the term 'wt' must be dimensionless, meaning it has no dimensions in terms of mass, length, and time.

An equation that is dimensionally consistent is not necessarily physically correct. For example, the equation of motion s = 2ut + at² is dimensionally correct, but it is physically incorrect because the numerical constants are wrong, as the actual equation is s = ut + 1/2 at². Dimensional analysis can only verify dimensional balance; it cannot verify the actual numerical magnitude of an equation or the correctness of the constants present in it. Refractive index is a dimensionless quantity. Although it involves the speed of light, it is the ratio of two identical quantities (speed of light in medium 1 / speed of light in medium 2), due to which its dimensions cancel each other out.

Key Points

According to the Principle of Homogeneity, the dimensions of each term on both sides of a physical equation must be the same. Only physical quantities with identical dimensions can be added to, subtracted from, or compared with one another. A major limitation of dimensional analysis is that it gives no information about pure numbers or constants (such as pi, 2) because they are dimensionless.

Additional Facts

If an equation is found to be dimensionally incorrect or inconsistent, it is definitely physically incorrect as well. Dimensional analysis is widely used to establish relationships between different physical quantities, convert units from one system to another, and perform preliminary checks on the correctness of equations. However, this method does not work for equations where a quantity depends on more than three fundamental quantities (mass, length, time).

Important Questions and Answers

Question: If the dimensions on both sides of an equation are equal, can that equation be considered one hundred percent physically correct?

Answer: No, because despite being dimensionally correct, the equation can be physically incorrect due to errors in numerical constants.

Question: What is the dimensional formula for the refractive index of a medium?

Answer: It is a dimensionless and unitless quantity, where all three fundamental dimensions (mass, length, time) are zero.