Import Question JSON

Current Question (ID: 9710)

Question:

$\text{A string of negligible mass going over a clamped pulley of mass } m \text{ supports a block of mass } M \text{ as shown in the figure. The force on the pulley by the clamp is given by:}$

Options:

1. $\sqrt{2Mg}$
2. $\sqrt{2mg}$
3. $g\sqrt{(M + m)^2 + m^2}$
4. $g\sqrt{(M + m)^2 + M^2}$

Solution:

\text{Force on the pulley by the clamp:} \text{From the force diagram, we can see that the pulley experiences:} \text{• Tension } T \text{ in the string (horizontal component)} \text{• Weight } mg \text{ of the pulley (downward)} \text{• Weight } Mg \text{ of the hanging block (transmitted through string tension)} \text{The resultant force that the clamp must provide is:} F_{pc} = \sqrt{T^2 + [(M + m)g]^2} \text{Since the system is in equilibrium, the tension } T = Mg \text{Substituting:} F_{pc} = \sqrt{(Mg)^2 + [(M + m)g]^2} \text{Factoring out } g\text{:} F_{pc} = g\sqrt{M^2 + (M + m)^2} \text{Therefore, the force on the pulley by the clamp is } g\sqrt{(M + m)^2 + M^2}

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
}