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A particle executes simple harmonic motion with an amplitude of 4 cm4 cm. At the mean position, the velocity of the particle is 10 cm s−110 cm s-1. The distance of the particle from the mean position when its speed becomes 5 cm s−15 cm s-1 is

Question

A particle executes simple harmonic motion with an amplitude of 4 cm4 cm. At the mean position, the velocity of the particle is 10 cm s−110 cm s-1. The distance of the particle from the mean position when its speed becomes 5 cm s−15 cm s-1 is

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Solution

In simple harmonic motion, the velocity v of a particle is given by the equation:

v = ω√(A² - x²)

where:

  • ω is the angular frequency,
  • A is the amplitude, and
  • x is the displacement from the mean position.

Given that the amplitude A is 4 cm and the velocity at the mean position (x = 0) is 10 cm/s, we can find ω using the equation:

v = ωA 10 cm/s = ω * 4 cm ω = 10 cm/s / 4 cm = 2.5 s⁻¹

Now, we want to find the displacement x when the velocity v is 5 cm/s. We can use the velocity equation:

5 cm/s = 2.5 s⁻¹ * √(4 cm)² - x² 5 cm/s = 2.5 s⁻¹ * √16 cm² - x² 2 cm/s = √16 cm² - x² 4 cm²/s² = 16 cm² - x² x² = 16 cm² - 4 cm² x² = 12 cm² x = √12 cm = 2√3 cm

So, the distance of the particle from the mean position when its speed becomes 5 cm/s is 2√3 cm.

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