Question 137: Units , Dimensions and Error Analysis

Question

Consider the efficiency of carnot's engine is given by $\eta=\frac{\alpha \beta}{\sin 	heta} \log_e \frac{\beta x}{k T}$, where $\alpha$ and $\beta$ are constants. If T is temperature, k is Boltzmann constant, $	heta$ is angular displacement and x has the dimensions of length. Then, choose the incorrect option :

Accepted Answer

Since, dimensions trigonometric function and logarithmic function are dimensionless quantities.
$
	herefore[\eta]=\left[\mathrm{M}^0 \mathrm{~L}^0 \mathrm{~T}^0ight]
$
Also, dimensions of temperature, $[T]=\left[\mathrm{M}^0 \mathrm{~L}^0 \mathrm{~T}{ }^0 \mathrm{~K}ight]$
Dimensions of Boltzmann constant, $[k]=\left[\mathrm{ML}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}ight]$
Dimension of $x=\left[\mathrm{M}^0 \mathrm{LT}^0ight]$
(A) $[\beta ] = \left[ {{{kT} \over x}} ight] = \left[ {{E \over x}} ight] = [ML{T^{ - 2}}] = [F]$
(B) $[\alpha \beta ] = [{M^0}{L^0}{T^0}]$
${[\alpha ]^{ - 1}} = [\beta ] = \left[ {{{kT} \over x}} ight]$
So ${[\alpha ]^{ - 1}}[x] = [kT] = [M{L^2}{T^{ - 2}}]$
(C) $\eta \sin 	heta  = \alpha \beta $
So $[\eta \sin 	heta ] = [\alpha \beta ]$
$[\eta ] = [{M^0}{L^0}{T^0}]$ it is dimensionless quantity
(D) $[\alpha ] 
e [\beta ]$

Answer

Dimensions of $\beta$ is same as that of force.

Answer

Dimensions of $\alpha^{-1} x$ is same as that of energy.

Answer

Dimensions of $\eta^{-1} \sin 	heta$ is same as that of $\alpha \beta$.

Answer

Dimensions of $\alpha$ is same as that of $\beta$.

Units , Dimensions and Error Analysis

Choose the correct answer: