Import Question JSON

Current Question (ID: 9759)

Question:

$\text{A block of mass } m \text{ lying on a rough horizontal plane is acted upon by a horizontal force } P \text{ and another force } Q \text{ inclined at an angle } \theta \text{ to the vertical. The block will remain in equilibrium if the coefficient of friction between it and the surface is:}$

Options:

1. $\frac{(P+Q\sin\theta)}{(mg+Q\cos\theta)}$
2. $\frac{(P\cos\theta+Q)}{(mg-Q\sin\theta)}$
3. $\frac{(P+Q\cos\theta)}{(mg+Q\sin\theta)}$
4. $\frac{(P\sin\theta-Q)}{(mg-Q\cos\theta)}$

Solution:

$\text{Hint: } f = \mu N$ $\text{Step: Find the coefficient of friction between the block and the surface.}$ $\text{By drawing the free body diagram of the block for critical condition}$ $\text{The normal reaction of the given block is:}$ $N = mg + Q\cos\theta$ $\text{The frictional force of the given block is:}$ $f = P + Q\sin\theta$ $\text{For the condition of equilibrium, } f \leq \mu N$ $\mu \geq \frac{f}{N}$ $\Rightarrow \mu \geq \frac{P+Q\sin\theta}{mg+Q\cos\theta}$ $\text{Hence, option (1) is the correct answer.}$

Import JSON File

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
}