Question

The couple per unit twist C is related to the rigidity modulus n, radius of the wire r and length of the wire l according to the equation $C = K{n^x}{r^y}{l^z}$ where K is dimensionless constant. The values of x, y and z respectively are:
A. 1, 1, 1.
B. 2, 4, 1.
C. 1, -4, 2.
D. 1, 4, -1.

Answer 1

Hint: By using the principle of homogeneity of dimensions, the form of expressions can be obtained if we know the factors upon which that quantity depends. The dimensional formula of C is $\left[ {{M^1}{L^2}{T^{ - 2}}} \right]$, rigidity modulus, n = $\left[ {{M^1}{L^{ - 1}}{T^{ - 2}}} \right]$, radius, r = $\left[ {{L^1}} \right]$ and length, l = $\left[ {{L^1}} \right]$

Complete answer:
According to the question,
$ \Rightarrow C = K{n^x}{r^y}{l^z}$ ….. (i)
Where K is a dimensionless quantity.
Substituting the dimensions of the quantities in equation (i),
$ \Rightarrow \left[ {{M^1}{L^2}{T^{ - 2}}} \right] = {\left[ {{M^1}{L^{ - 1}}{T^{ - 2}}} \right]^x}{\left[ {{L^1}} \right]^y}{\left[ {{L^1}} \right]^z}$
Simplifying this,
$ \Rightarrow \left[ {{M^1}{L^2}{T^{ - 2}}} \right] = \left[ {{M^x}{L^{ - x + y + z}}{T^{ - 2x}}} \right]$
Comparing both sides, we get
$ \Rightarrow x = 1$ ……. (ii)
$ \Rightarrow 2 = - x + y + z$ ……. (iii)
From equation (ii) and (iii), we will get
$ \Rightarrow x = 1,y = 4,z = - 1$

So, the correct answer is “Option D”.

Note:
Whenever we ask such types of questions, first we have to identify all the quantities that are involved in the given equation. Then we will use the dimensional formula of those quantities and put that in the equation. After that, by equating and solving that equation, we will get the required value of unknown terms.