Question

Y = A sin ( ωt - kx ). Write the dimensions of w and k if x is distance and t is time.

Answer 1

Step 1: Given

Dimension is the minimum number of coordinates required to specify any place within it.

We have to calculate dimension for,

$\mathrm{Y}=\mathrm{A}\sin (\mathrm{\omega t}-\mathrm{kx})$

Step 2: Calculating dimensions for $\omega$

$$\begin{gathered} Y=A\sin (\omega t-kx)---\left(i\right) \\ Y=A\sin \theta ---\left(ii\right) \end{gathered}$$

By comparing equations (i) and (ii), we get

$\theta =(\omega t-kx)$

By the principle of homogeneity of dimensions,

$\left[LHS\right]=\left[RHS\right]$

$\Rightarrow \left[\theta \right]=\left[\omega t\right]=\left[kx\right]---\left(iii\right)$

Trigonometric quantities do not have dimensions so,

$$\begin{gathered} \left[\theta \right]=\left[M^0{L}^0{T}^0\right] \\ \left[\omega t\right]=\left[M^0{L}^0{T}^0\right] \end{gathered}$$

$\left[\omega \right]·\left[t\right]=\left[M^0{L}^0{T}^0\right]$

Step 3: Calculating dimensions for $k$

From equation (iii),

$$\begin{gathered} \left[\theta \right]=\left[M^0{L}^0{T}^0\right] \\ \left[kx\right]=\left[\theta \right] \\ \left[k\right]·\left[x\right]=\left[\theta \right] \\ \left[k\right]=\frac{\left[\theta \right]}{\left[x\right]} \\ \left[k\right]=\frac{\left[M^0{L}^0{T}^0\right]}{\left[L\right]} \\ \left[k\right]=\left[M^0{L}^{-1}{T}^0\right] \end{gathered}$$

Dimensions of $\omega$ is $\left[M^0{L}^0{\mathrm{T}}^{-1}\right]$ and dimensions of $k$ is $\left[M^0{L}^{-1}{T}^0\right]$.