Question

Force $F = P + Qt + \dfrac{1}{r + xs} + c\sin(\omega t + \phi)$ where $x$ and $t$ represent displacement and time respectively. The dimensions of the product $cs$ are

• A. $\left[L^{1}M^{0}T^{0}\right]$
• B. $\left[L^{-1}M^{1}T^{0}\right]$
• C. $\left[L^{-1}M^{0}T^{0}\right]$
• D. $\left[L^{1}M^{-1}T^{1}\right]$

Hint:

In dimensional analysis, arguments of trigonometric functions must always be dimensionless.

Answer 1

The Correct Option is C

Step 1: Dimensional consistency of force equation.
All terms in the equation of force must have the same dimensions as force.
Step 2: Analyze sine term.
The argument of sine function must be dimensionless, hence $(\omega t + \phi)$ is dimensionless.
Step 3: Dimension of constant $c$.
Since $\sin(\omega t + \phi)$ is dimensionless, the dimension of $c$ must be same as force.
Step 4: Analyze the term $\dfrac{1{r + xs}$. For dimensional consistency, $r$ and $xs$ must have same dimensions.
Hence, dimension of $s$ is inverse of displacement, i.e., $[L^{-1}]$.
Step 5: Find dimensions of $cs$.
\[ [c] = [MLT^{-2}], \quad [s] = [L^{-1}] \] \[ [cs] = [M^{0}L^{-1}T^{0}] \]