Question

The velocity $\left(v\right)$ of a particle depends upon the time $t$ according to the equation
$v=ab{e}^{-bt}\cos \omega t$
where $a$ and $b$ are positive constants. Then the dimension of $b$ is

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

Answer 1

The correct option is D $\left[M{}^{\circ }L{}^{\circ }{T}^{-1}\right]$

As we know that, the power of an exponent is a number. Therefore, dimensionally, $bt=M^{\circ }{L}^{\circ }{T}^{\circ }$
The dimension of time $\left(t\right)$ is $\left[M{}^{\circ }L{}^{\circ }{T}^1\right]$
So the dimension of $b$ is$$\therefore b=\frac{\left[M{}^{\circ }L{}^{\circ }T{}^{\circ }\right]}{\left[T\right]}=\left[M{}^{\circ }L{}^{\circ }{T}^{-1}\right]$$

Why this question? Tip:- Constant such as $3,\pi$ or trigonometrical functions are dimensionless.