First slide

Accuracy; precision of instruments and errors in measurements

Question

The wavelength associated with a moving particle depends up on p^th power of its mass m, q^th power of its velocity v and r^th power of Planck's constant h. Then the correct set of values of p, q and r is

Moderate

A

$p = 1, q = - 1, r = 1$
B

$p = 1, q = 1, r = 1$
C

$p = - 1, q = - 1, r = - 1$
D

$p = - 1, q = - 1, r = 1$

Solution

Given $λ = k m^{p} v^{q} h^{k}$ The dimensions of right-hand rule and left-hand side terms should be equal.
$\begin{array}{l} So [M^{0} L T^{0}] = [M]^{p} {[L T^{- 1}]}^{q} {[M L^{2} T^{- 1}]}^{r} \\ or [M^{0} L T^{0}] = [M^{p + r}] [L^{q + 2 r}] [T^{- q - r}] \end{array}$
Now, compare powers of M, L and T, we get
$\begin{array}{l} p + r = 0 . . . . . . (i) \\ q + 2 r = 1 . . . . . . . (i i) \\ - q - r = 0 . . . . . . . (i i i) \end{array}$
After solving p= -1, q = -1 and r=1 putting these values, we get
$λ = k \frac{h}{m v} which is the required relation.$

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