Question

The displacement of a body at a particular second n is given by the expression ${S_n}^{th} = u + \left( {2n - 1} \right)$. The dimensional formula of \[{S_n}^{th}\] in this equation is
(A) $\left[ {{M^2}{L^0}{T^1}} \right]$.
(B) $\left[ {{M^0}{L^1}{T^0}} \right]$.
(C) $\left[ {{M^0}{L^1}{T^{ - 1}}} \right]$.
(D) $\left[ {{M^0}{L^0}{T^0}} \right]$.

Answer 1

Hint: In the given expression ${S_n}^{th} = u + \left( {2n - 1} \right)$, ${S_n}^{th}$ is represented as the displacement of the body at a particular second n. To find the dimensional formula of ${S_n}^{th}$ in the given equation, formula of displacement should be use, also by using the dimensions of all the symbols that used in the displacement equation for the nth second, we can find the dimensional formula of ${S_n}^{th}$ for the given equation.

Complete step by step answer: Given: ${S_n}^{th} = u + \left( {2n - 1} \right)$
${S_n}^{th}$ = displacement of the body at nth second .
a= acceleration of the body.
u= initial velocity of the body.
Now taking L.H.S, we get,
${S_n}^{th}$ $ = initial velocity \times time$$ = [{M^0}{L^1}{T^{ - 1}}] \times [{M^0}{L^0}{T^1}] = [{M^0}{L^1}{T^0}]$.
Now taking R.H.S , we get,
$RHS = \;u + \dfrac{a}{2}\left( {2n - 1} \right)$
u= initial velocity = $[{M^0}{L^1}{T^{ - 1}}]$
a=acceleration= $[{M^0}{L^1}{T^{ - 2}}]$
Time=(2n-1)=n= $[{M^0}{L^0}{T^1}]$
1/2 is the dimensionless quantity.
Therefore, R.H.S= $[{M^0}{L^1}{T^{ - 1}}]$$ + $$[{M^0}{L^1}{T^{ - 2}}]$$ \times $$[{M^0}{L^0}{T^1}]$$ = $$[{M^0}{L^1}{T^0}]$
Hence, L.H.S=R.H.S
Therefore, ${S_n}^{th}$ dimensional formula is $[{M^0}{L^1}{T^0}]$.
Hence, option (B) is the correct option.

Note: Whenever we come up with this type of problem, you must remember the formula that the displacement travelled by a body during n seconds is ${S_n}^{th} = \;u + \dfrac{a}{2}\left( {2n - 1} \right)$. Here, we first found the dimensional formula of the L.H.S of the equation, then found the dimensional formula R.H.S of the equation. After, equating LHS and RHS side we will get the dimensional formula of ${S_n}^{th}$. of the given equation.