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Current Question (ID: 11289)

Question:

$\text{A mass m is suspended from two springs of spring constant } \text{k}_1 \text{ and } \text{k}_2 \text{ as shown in the figure below. The time period of vertical oscillations of the mass will be}$

Options:

1. $2\pi \sqrt{\frac{\text{k}_1+\text{k}_2}{\text{m}}}$
2. $2\pi \sqrt{\frac{\text{m}}{\text{k}_1+\text{k}_2}}$
3. $2\pi \sqrt{\frac{\text{m}(\text{k}_1\text{k}_2)}{\text{k}_1+\text{k}_2}}$
4. $2\pi \sqrt{\frac{\text{m}(\text{k}_1+\text{k}_2)}{\text{k}_1\text{k}_2}}$ (Correct)

Solution:

$\text{The two springs are connected in series. The effective spring constant } \text{K}_{\text{eq}} \text{ for springs in series is given by the formula } \frac{1}{\text{K}_{\text{eq}}}=\frac{1}{\text{k}_1}+\frac{1}{\text{k}_2}. \text{Solving this, we get } \text{K}_{\text{eq}}=\frac{\text{k}_1\text{k}_2}{\text{k}_1+\text{k}_2}. \text{The time period of oscillation } \text{T} \text{ is given by the formula } \text{T}=2\pi \sqrt{\frac{\text{m}}{\text{K}_{\text{eq}}}}. \text{Substituting the value of } \text{K}_{\text{eq}}, \text{ we get } \text{T}=2\pi \sqrt{\frac{\text{m}}{\frac{\text{k}_1\text{k}_2}{\text{k}_1+\text{k}_2}}}=2\pi \sqrt{\frac{\text{m}(\text{k}_1+\text{k}_2)}{\text{k}_1\text{k}_2}}.$

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
}