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Simple Harmonic Motion question

2024 · 29 Jan · Shift 2 · Q75
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Simple Harmonic Motion question

2024 · 29 Jan · Shift 2 · Q75

JEE MainPhysicsSimple Harmonic MotionMCQ+4 / −1
The bob of a pendulum was released from a horizontal position. The length of the pendulum is 10 m10 \mathrm{~m}10 m. If it dissipates 10%10 \%10% of its initial energy against air resistance, the speed with which the bob arrives at the lowest point is: [Use, g:10 ms−2\mathrm{g}: 10 \mathrm{~ms}^{-2}g:10 ms−2]
  1. A
    56 ms−15 \sqrt{6} \mathrm{~ms}^{-1}56​ ms−1
  2. B
    55 ms−15 \sqrt{5} \mathrm{~ms}^{-1}55​ ms−1
  3. C
    25 ms−12 \sqrt{5} \mathrm{~ms}^{-1}25​ ms−1
  4. D
    65 ms−16 \sqrt{5} \mathrm{~ms}^{-1}65​ ms−1
View written solutionFree

Correct answer: D

  1. Initial position and energy

The pendulum bob is released from the horizontal position.

So initially, the bob is at the same height as the point of suspension, while at the lowest point it is vertically below the suspension point by a distance equal to the length of the pendulum.

Hence the vertical drop is h=l=10 m.h = l = 10\,\text{m}.h=l=10m.

Therefore, the initial potential energy relative to the lowest point is Ei=mgh=m⋅10⋅10=100m.E_i = mgh = m \cdot 10 \cdot 10 = 100m.Ei​=mgh=m⋅10⋅10=100m.

  1. Energy dissipated

It is given that 10%10\%10% of the initial energy is dissipated against air resistance.

So the remaining energy at the lowest point is E=90% of Ei=0.9×100m=90m.E = 90\% \text{ of } E_i = 0.9 \times 100m = 90m.E=90% of Ei​=0.9×100m=90m.

  1. Convert remaining energy into kinetic energy

At the lowest point, this remaining energy is kinetic: 12mv2=90m.\frac{1}{2}mv^2 = 90m.21​mv2=90m.

Cancel mmm: 12v2=90\frac{1}{2}v^2 = 9021​v2=90 v2=180v^2 = 180v2=180 v=180=36⋅5=65 m/s.v = \sqrt{180} = \sqrt{36 \cdot 5} = 6\sqrt{5}\,\text{m/s}.v=180​=36⋅5​=65​m/s.

  1. Match with options

v=65 m/sv = 6\sqrt{5}\,\text{m/s}v=65​m/s which corresponds to Option D.


Verification with stored answer: Stored correct answer is D, which matches our result.

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