Question 83: Units , Dimensions and Error Analysis

Question

The equation of stationary wave is :
$$y=2 \mathrm{a} \sin \left(\frac{2 \pi \mathrm{nt}}{\lambda}ight) \cos \left(\frac{2 \pi x}{\lambda}ight) 	ext {. }$$
Which of the following is NOT correct :

Accepted Answer

To determine which of the options is NOT correct, we need to analyze the dimensional consistency of each term in the given equation of the stationary wave:
$$y = 2a \sin \left(\frac{2\pi nt}{\lambda}ight) \cos \left(\frac{2\pi x}{\lambda}ight).$$
Let's break down the dimensions for each relevant term:
1. Analyzing $$\mathrm{nt}$$:
The argument of the sine function $$\frac{2\pi nt}{\lambda}$$ must be dimensionless. Therefore, the dimensions of $$\mathrm{nt}$$ should be the same as the dimensions of $$\lambda$$ (wavelength), which is [L].
Thus, the dimensions of $$nt$$ should be [L].
Hence, Option A is correct.
2. Analyzing $$n$$:
From the above analysis, since $$\mathrm{nt}$$ has the dimension [L] and $$t$$ (time) has the dimension [T], it follows that:
$$n = \frac{	ext{Dimension of } nt}{	ext{Dimension of } t} = \frac{[L]}{[T]} = [\mathrm{LT}^{-1}].$$
Hence, Option B is also correct.
3. Analyzing $$x$$:
In the argument of the cosine function $$\frac{2\pi x}{\lambda}$$, since it must be dimensionless, the dimensions of $$x$$ should be the same as the dimensions of $$\lambda$$ (wavelength), which is [L].
Hence, the dimensions of $$x$$ should be [L].
Therefore, Option C is correct.
4. Analyzing $$\frac{n}{\lambda}$$:
From the dimensions we have determined for $$n$$ and $$\lambda$$:
$$\frac{n}{\lambda} = \frac{[\mathrm{LT}^{-1}]}{[L]} = [\mathrm{T}^{-1}].$$
Thus, the dimensions of $$\frac{n}{\lambda}$$ should be [T
-1
], not [T].
This indicates that Option D is NOT correct.
Conclusion:
Option D is the correct answer since it is NOT dimensionally correct.

Answer

The dimensions of $$\mathrm{nt}$$ is [L]

Answer

The dimensions of $$n$$ is $$[\mathrm{LT}^{-1}]$$

Answer

The dimensions of $$x$$ is [L]

Answer

The dimensions of $$n / \lambda$$ is [T]

Units , Dimensions and Error Analysis

The equation of stationary wave is : $y=2 \mathrm{a} \sin \left(\frac{2 \pi \mathrm{nt}}{\lambda}\right) \cos \left(\frac{2 \pi x}{\lambda}\right) \text {. }$ Which of the following is NOT correct :

Choose the correct answer: