Question

The wavelength associated with $a$ moving particle depends upon $p^{th}$ power of its mass $m, q^{th}$ power of its velocity $v$ and rth power of Planck’s constant h. Then the sum of values of $p, q$ and $r$ is $-K.$ Find the value of $K$?

Answer 1

Given $\lambda= km^{p} v^{p} h^{k}$. The dimensions of right-hand rule and left-hand side terms should be equal.
So $[M^{0} LT^{0}]=[M]^{p} [LT^{-1}]^{q} [ML^{2}T^{-1}]^{r}$
or $[M^{0} LT^{0}] =[M^{p+r}][L^{q+2r}][T^{-q-r}]$
Now, compare powers of $M, L$ and $T,$ we get
$p + r= 0 \,\,\,\, ...(i)$
$q + 2r=1 \,\,\,\, ...(ii)$
$- q - r = 0\,\,\, ...(iii)$
After solving p $= -1 , q = -1$ and $r = 1$ putting these values, we get