Question

Find the dimensions of \[A \times (B + C) \times D\]where \[A\]is mass, dimensions of \[B\]is unknown, \[C\]is energy, and \[D\] is dimensionless constant:
A. \[{M^2}{L^2}{T^{ - 1}}\]
B. \[{M^2}{L^2}{T^{ - 2}}\]
C. \[{M^2}{L^1}{T^{ - 2}}\]
D. \[{M^1}{L^2}{T^{ - 2}}\]

Answer 1

Hint: In this question, we have to find the dimension of \[A \times (B + C) \times D\], where \[A\] is mass, dimensions of \[B\] is unknown, \[C\]is energy, and \[D\] is dimensionless constant. We know that dimension of energy in terms of mass, length and time is given by \[{M^1}{L^2}{T^{ - 2}}\]. We will be using the Homogeneity principle of dimensional analysis to find the dimension of \[A \times (B + C) \times D\]. Homogeneity principle states that the dimensions are similar for every equation that represents a physical situation.


Complete step by step solution:
Let us consider that \[{\mathbf{B}}\] has the same dimension as \[C\].
Therefore, the dimension of \[{\mathbf{B}}\] is \[{M^1}{L^2}{T^{ - 2}}\].
Mathematically, the dimension of \[B + C\] is the sum of dimensions of \[{\mathbf{B}}\& C\] i.e.,
Dimension of \[B + C = \] Dimension of \[B + \] Dimension of \[C\]
Dimension of \[B + C = {M^1}{L^2}{T^{ - 2}} + {M^1}{L^2}{T^{ - 2}}\]
Dimension of \[B + C = 2 \times {M^1}{L^2}{T^{ - 2}} \]
Thus, the dimension of \[B + C\] is \[{M^1}{L^2}{T^{ - 2}}\].
\[D\] has no dimension and \[A\] has dimension \[M\].
The dimension of \[A \times (B + C) \times D\] will be the product of dimensions of \[A\], \[B + C\] and \[D\].
Thus, dimension of \[A \times (B + C) \times D = \] Dimension of \[A \times \]Dimension of \[\left( {B + C} \right) \times \] Dimension of \[D\]
Dimension of \[A \times (B + C) \times D = M \times {M^1}{L^2}{T^{ - 2}} = {M^2}{L^2}{T^{ - 2}}\].

Therefore, the correct option is B. .

Additional Information : The dimension confines used in mechanics are time, mass, and length. Symbolically, these are written as t, m, and l, independently. The study of electromagnetism adds a fresh abecedarian dimension, electric charge, or q. Other amounts have confines compounded of these.

Note: Students may make mistakes while defining the required dimensions. Some of the students may write dimensions as \[{M^1}{L^2}{T^{ - 2}}\]in a hurry but this is not right. Students must write all the dimensions correctly without hurry.