Import Question JSON

Current Question (ID: 9730)

Question:

$\text{A particle is on a smooth horizontal plane. A force } F \text{ is applied, whose } (F-t) \text{ graph is given.}$ $\text{Consider the following statements:}$ $\text{(a) At time } t_1\text{, acceleration is constant.}$ $\text{(b) Initially the particle must be at rest.}$ $\text{(c) At time } t_2\text{, acceleration is constant.}$ $\text{(d) The initial acceleration is zero.}$ $\text{Select the correct statement(s):}$

Options:

1. $\text{(a), (c)}$
2. $\text{(a), (b), (d)}$
3. $\text{(c), (d)}$
4. $\text{(b), (c)}$

Solution:

$\text{Hint: } \vec{a} = \frac{\vec{F}}{m}$ $\text{Step 1: Analyze motion in time interval 0 to } t_0\text{.}$ $F \propto t$ $a \propto t$ $\text{At } t = 0\text{, } F = 0$ $\text{The particle may or may not be at rest.}$ $\text{Step 2: Analyze motion after time } t_0\text{.}$ $\text{The force is constant at the time } t_2 \text{ and hence the acceleration.}$ $\text{Analysis of each statement:}$ $\text{(a) At time } t_1\text{, acceleration is constant - FALSE. During } 0 \text{ to } t_1\text{, force varies linearly with time, so acceleration also varies.}$ $\text{(b) Initially the particle must be at rest - FALSE. At } t = 0\text{, } F = 0 \text{ but this doesn't mean the particle is at rest; it could have initial velocity.}$ $\text{(c) At time } t_2\text{, acceleration is constant - TRUE. After } t_1\text{, force is constant, so acceleration is constant.}$ $\text{(d) The initial acceleration is zero - TRUE. At } t = 0\text{, } F = 0\text{, so } a = F/m = 0\text{.}$

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
}