The efficiency of an engine is given by eta = | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Efficiency $\eta$ is a dimensionless quantity. The argument of the logarithmic function must also be dimensionless, so $\frac{\beta x}{kT} = 1$ (d

Vedclass · Accepted Answer

Dimensions of $\alpha$ are same as that of $\beta$

Vedclass · Answer

(D) Efficiency $\eta$ is a dimensionless quantity. The argument of the logarithmic function must also be dimensionless, so $\frac{\beta x}{kT} = 1$ (dimensionless).Since $\eta$ is dimensionless, $[\eta] = [M^0 L^0 T^0]$.From $\frac{\beta x}{kT} = 1$, we get $\beta = \frac{kT}{x}$.Given $k = [M L^2 T^{-2} K^{-1}]$ and $T = [K]$, we have $\beta = \frac{[M L^2 T^{-2} K^{-1}][K]}{[L]} = [M L T^{-2}]$.This is the dimension of force, so option $A$ is correct.Since $\eta = \frac{\alpha \beta}{\sin 	heta} \cdot 	ext{constant}$, and $\eta$ and $\sin 	heta$ are dimensionless, $[\alpha \beta] = [1]$, which implies $[\alpha] = [\beta^{-1}] = [M^{-1} L^{-1} T^2]$.Option $D$ states $[\alpha] = [\beta]$, which is incorrect.For option $B$, $[\alpha^{-1} x] = [\beta x] = [M L T^{-2}][L] = [M L^2 T^{-2}]$, which is the dimension of energy. Thus, $B$ is correct.For option $C$, $[\eta^{-1} \sin 	heta] = [1] = [\alpha \beta]$, which is correct.

Similar Questions

The density of a material in $SI$ units is $128 \ kg \ m^{-3}$. In a system of units where the unit of length is $25 \ cm$ and the unit of mass is $50 \ g$,the numerical value of the density of the material is:

If the units of force,energy,and velocity are respectively $10\, N$,$100\, J$,and $5\, m/s$,then the units of length,mass,and time will be:

If $C$ (the velocity of light), $h$ (Planck's constant), and $G$ (gravitational constant) are taken as fundamental quantities, then the dimensional formula of mass is

In the relation $P = \frac{\alpha}{\beta} e^{-\frac{\alpha Z}{k\theta}}$,$P$ is pressure,$Z$ is distance,$k$ is Boltzmann constant,and $\theta$ is temperature. The dimensional formula of $\beta$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module