Question 122: 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: