Question

The Force $F$ is given as $F = at + b{t^2}$ where $t$ is time. What are the dimensions of $a$ and $b$
A. $[M][L]{[T]^{ - 1}}$ and $[M][L]{[T]^0}$
B. $[M][L]{[T]^{ - 3}}$ and $[M]{[L]^2}{[T]^4}$
C. $[M][L]{[T]^{ - 4}}$ and $[M][L]{[T]^1}$
D. $[M][L]{[T]^{ - 3}}$ and $[M][L]{[T]^{ - 4}}$

Answer 1

Hint: Only quantities with similar dimensions can be equated or added. The dimension of Force is ${[M]^1}{[L]^1}{[T]^{ - 2}}$ and that of time is ${[M]^0}{[L]^0}{[T]^1}$. The dimension of $at$ and that of $b{t^2}$ should have the same dimension as that of force.

Complete step-by-step answer:
Equations of any kind have to satisfy dimensional consistency. This means that all individual terms in an equation should have the same dimension. For example, if you have an equation $AB = C + \dfrac{D}{E}$, the dimensions of $AB$, $C$ and $\dfrac{D}{E}$ should be equal.
This is because objects with the same dimensions can only be equated, added or subtracted. It makes no sense to subtract $2cm$ from $3kg$

Here in the question, $F$ is given as $at + b{t^2}$, where t is time and has the dimension: ${[M]^0}{[L]^0}{[T]^1}$
Now we know from our discussion that $F$, $at$, and $b{t^2}$ should have the same dimension.
The dimension of $F$ is ${[M]^1}{[L]^1}{[T]^{ - 2}}$
So the dimension of $at$ should also be ${[M]^1}{[L]^1}{[T]^{ - 2}}$
substituting the dimension of t as ${[M]^0}{[L]^0}{[T]^1}$, we get
$[a] \times {[M]^0}{[L]^0}{[T]^1} = {[M]^1}{[L]^1}{[T]^{ - 2}}$
So we get $[a] = {[M]^1}{[L]^1}{[T]^{ - 3}}$

Similarly, for b,
The dimension of $b{t^2}$ should also be ${[M]^1}{[L]^1}{[T]^{ - 2}}$.
$[b] \times {\left( {{{[M]}^0}{{[L]}^0}{{[T]}^1}} \right)^2} = {[M]^1}{[L]^1}{[T]^{ - 2}}$
$[b] \times {[M]^0}{[L]^0}{[T]^2} = {[M]^1}{[L]^1}{[T]^{ - 2}}$
So we get $[b] = {[M]^1}{[L]^1}{[T]^{ - 4}}$
These answers correspond to option D and hence, it is the right answer.

Additional Information:
Dimensions of commonly used quantities are listed below. Remembering them would be beneficial.
Force ${[M]^1}{[L]^1}{[T]^{ - 2}}$
Work ${[M]^1}{[L]^2}{[T]^{ - 2}}$
spring constant ${[M]^1}{[T]^{ - 2}}$
Moment of Inertia ${[M]^1}{[L]^2}$
Electric Field ${[M]^1}{[L]^1}{[T]^{ - 3}}{[A]^ {- 1}}$
Charge ${[T]^1}{[A]^1}$

Note: Though a correct equation should be dimensionally consistent. A dimensionally consistent equation need not be correct. Factors like the presence or absence of dimensionless constants, logarithms, sines, etc cannot be verified using dimensional analysis.