$[\text{M}^0\text{L}^1\text{T}^{-2}]$
The problem asks for the dimensional formula of the constant 'b' in the equation relating distance '$s$' and time '$t$':
$ s = at + bt^2 $
Dimensional Homogeneity Principle
According to the principle of dimensional homogeneity, every term in a physically consistent equation must have the same dimensions. We know the dimensions of distance '$s$' and time '$t$':
- Dimension of distance ($s$): $[s] = [\text{L}^1]$
- Dimension of time ($t$): $[t] = [\text{T}^1]$
Dimensional Formula of 'a'
Equating the dimensions of '$s$' and '$at$':
$ [s] = [a][t] $
$ [\text{L}^1] = [a][\text{T}^1] $
Solving for the dimensions of 'a':
$ [a] = \frac{[\text{L}^1]}{[\text{T}^1]} = [\text{L}^1\text{T}^{-1}] $
Dimensional Formula of 'b'
Equating the dimensions of '$s$' and '$bt^2$':
$ [s] = [b][t^2] $
$ [\text{L}^1] = [b][\text{T}^1]^2 $
$ [\text{L}^1] = [b][\text{T}^2] $
Solving for the dimensions of 'b':
$ [b] = \frac{[\text{L}^1]}{[\text{T}^2]} = [\text{L}^1\text{T}^{-2}] $
In terms of mass [M], length [L], and time [T], the dimensional formula is $[\text{M}^0\text{L}^1\text{T}^{-2}]$.
Similar Questions
- The dimensional formula of force:
- Which of the following is dimensionless?
- The dimension of surface tension is ______.
Important Questions from Dimensions of physical quantities
The dimensional formula of force:
- Which of the following combinations of fundamental constants has the dimension of length, $[L^1]$? (Given: Planck constant $h = [ML^2T^{-1}]$, speed of light $c = [LT^{-1}]$, gravitational constant $G = [M^{-1}L^3T^{-2}])$
What is the formula of velocity gradient?
- The dimensional formula of force:
- Which of the following is dimensionless?
