The relative density of an alloy is 6.5. Find the density of the alloy. (Take density of water = 1 g/cm³). (a) 12. 1 g/cm³ (b) 6.5 g/cm³ (c) 8.0 g/cm³ (d) 5.5 g/cm³

Relative density is defined as the ratio of the density of a substance to the density of a reference substance (usually water).

We are given the relative density of the alloy as 6.5 and the density of water as $1 \text{ g/cm}^3$.

The calculated density matches option (b).

The final answer is $\boxed{\text{b) } 6.5 \text{ g/cm}^3}$.

• The dimension of velocity ($v$ or $u$) is $[L T^{-1}]$. Therefore, the dimension of $v^2$ or $u^2$ is $[L T^{-1}]^2 = [L^2 T^{-2}]$.
• The dimension of acceleration ($a$) is $[L T^{-2}]$.
• The dimension of displacement ($s$) is $[L]$.
• The number 2 is a dimensionless constant.

• Dimension of $v^2$: $[L^2 T^{-2}]$
• Dimension of $u^2$: $[L^2 T^{-2}]$
• Dimension of $2as$: $[L T^{-2}] \times [L] = [L^2 T^{-2}]$

All terms in the equation ($v^2$, $u^2$, and $2as$) have the same dimension, which is $[L^2 T^{-2}]$.

Since the dimensions of all terms on both sides of the equation are the same, the equation is dimensionally valid.

The final answer is $\boxed{\text{b) valid}}$.