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Current Question (ID: 8738)

Question:

$\text{If } y = a \sin(bt - cx)\text{, where } y \text{ and } x \text{ represent length and } t \text{ represents time, then which of the following has the same dimensions as that of } \frac{ab^2}{c}\text{?}$

Options:

1. $\text{(speed)}^2$
2. $\text{momentum}$
3. $\text{angle}$
4. $\text{acceleration}$

Solution:

$\text{Hint: Find the dimension of constant.}$ $\text{Step 1: Find the dimensions of } a\text{, } b\text{, and } c\text{.}$ $y = a \sin(bt - cx)$ $\text{Since the argument of sine function must be dimensionless:}$ $[bt] = [M^0L^0T^0] \Rightarrow [b] = [M^0L^0T^{-1}] = [T^{-1}]$ $[cx] = [M^0L^0T^0] \Rightarrow [c] = [M^0L^{-1}T^0] = [L^{-1}]$ $\text{Since } y \text{ represents length: } [a] = [L]$ $\text{Step 2: Find the dimension of the given expression.}$ $\Rightarrow \left[\frac{ab^2}{c}\right] = \frac{[L][T^{-1}]^2}{[L^{-1}]} = \frac{[L][T^{-2}]}{[L^{-1}]} = [LT^{-1}]^2$ $\text{Therefore, the dimensions of the given expression are } [LT^{-1}]^2\text{.}$ $\text{This is equivalent to (speed)}^2 \text{ since speed has dimensions } [LT^{-1}]\text{.}$ $\text{Hence, option (1) is the correct answer.}$

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Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}