Question

The displacement of a particle moving along the x-axis with respect to time t is\[x = at + b{t^2} - c{t^3}\]. Then find the dimensions of c.
A. \[\left[ {{T^{ - 3}}} \right]\]
B. \[\left[ {L{T^{ - 2}}} \right]\]
C. \[\left[ {L{T^{ - 3}}} \right]\]
D. \[\left[ {L{T^3}} \right]\]
E. \[\left[ {L{T^2}} \right]\]

Answer 1

Hint: An equation, which gives the relation between the fundamental units and derived units in terms of dimensions is known as dimensional formula or equation. Usually, in mechanics, there are three base dimensions, which are length, mass, and time and are represented by the letters L, M, and T respectively.

Complete step by step solution:
Consider the equation, \[x = at + b{t^2} - c{t^3}\]
Now, writing the dimensional formula on both sides we get,
Dimension of x = Dimension of \[c{t^3}\]
\[\left[ L \right] = \left[ c \right]\left[ {{T^3}} \right] \\ \]
\[\Rightarrow \left[ c \right] = \dfrac{{\left[ L \right]}}{{\left[ {{T^3}} \right]}} \\ \]
\[\therefore \left[ c \right] = \left[ {L{T^{ - 3}}} \right] \\ \]
Therefore, the dimension of c is \[\left[ {L{T^{ - 3}}} \right]\].

Hence, option C is the correct answer.

Additional informationDimensionless quantities are those which do not have any dimensions but have a fixed value. These dimensional quantities are of two types;
1. Dimensionless quantities without units: Pure numbers and the trigonometric functions are the dimensionless quantities without units
2. Dimensionless quantities with units: Angular displacement, Joule’s constant, etc are the quantities having units.

Note: While solving this problem it is important to know the dimensional formula of certain terms and we need to equate the left-hand side and the right-hand side dimensional formula and both should be equal.