Question

In two systems of relations among velocity, acceleration and force are respectively $v _ { 2 } = \frac { \alpha ^ { 2 } } { \beta } v _ { 1 }$ $a_{2} = \alpha\beta a_{1}$ and $F_{2} = \frac{F_{1}}{\alpha\beta}.$ If $\alpha$ and $\beta$are constants then relations among mass, length and time in two systems are

• A. $M_{2} = \frac{\alpha}{\beta}M_{1},L_{2} = \frac{\alpha^{2}}{\beta^{2}}L_{1},T_{2} = \frac{\alpha^{3}T_{1}}{\beta}$
• B. $M_{2} = \frac{1}{\alpha^{2}\beta^{2}}M_{1},L_{2} = \frac{\alpha^{3}}{\beta^{3}}L_{1},T_{2} = T_{1}\frac{\alpha}{\beta^{2}}$
• C. $M_{2} = \frac{\alpha^{3}}{\beta^{3}}M_{1},L_{2} = \frac{\alpha^{2}}{\beta^{2}}L_{1},T_{2} = \frac{\alpha}{\beta}T_{1}$
• D. $M_{2} = \frac{\alpha^{2}}{\beta^{2}}M_{1},L_{2} = \frac{\alpha}{\beta^{2}}L_{1},T_{2} = \frac{\alpha^{3}}{\beta^{3}}T_{1}$

Answer 1

Answer: (B)

$M_{2} = \frac{1}{\alpha^{2}\beta^{2}}M_{1},L_{2} = \frac{\alpha^{3}}{\beta^{3}}L_{1},T_{2} = T_{1}\frac{\alpha}{\beta^{2}}$

Answer 2

$v_{2} = v_{1}\frac{\alpha^{2}}{\beta}$ $\Rightarrow \lbrack L_{2}T_{2}^{- 1}\rbrack = \lbrack L_{1}T_{1}^{- 1}\rbrack\frac{\alpha^{2}}{\beta}$ ......(i)

$a_{2} = a_{1}\alpha\beta$ $\Rightarrow \lbrack L_{2}T_{2}^{- 2}\rbrack = \lbrack L_{1}T_{1}^{- 2}\rbrack\alpha\beta$ ......(ii)

and $F_{2} = \frac{F_{1}}{\alpha\beta}$ $\Rightarrow \lbrack M_{2}L_{2}T_{2}^{- 2}\rbrack = \lbrack M_{1}L_{1}T_{1}^{- 2}\rbrack \times \frac{1}{\alpha\beta}$......(iii)

Dividing equation (iii) by equation (ii) we get

$M_{2} = \frac{M_{1}}{(\alpha\beta)\alpha\beta}$ $= \frac{M_{1}}{\alpha^{2}B^{2}}$

Squaring equation (i) and dividing by equation (ii) we get $L_{2} = L_{1}\frac{\alpha^{3}}{\beta^{3}}$

Dividing equation (i) by equation (ii) we get $T_{2} = T_{1}\frac{\alpha}{\beta^{2}}$