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calculate-kinetic-energy-in-shm

๐Ÿš€ In simple harmonic motion (SHM), an object oscillates about an equilibrium position. The energy in SHM can be divided into kinetic energy (KE) and potential energy (PE). The kinetic energy of an object in SHM can be calculated using the formula KE = (1/2)mvยฒ, where m is the mass of the object and v is its velocity. In SHM, the velocity varies as the object moves, reaching maximum at the equilibrium position and zero at the extreme positions. The total mechanical energy in SHM remains constant and is the sum of kinetic and potential energy at any point in time.

Theory Explanation

Understanding Velocity in SHM

In SHM, the velocity of the oscillating object changes continuously. At the maximum displacement (amplitude), the velocity is zero, and at the equilibrium position, the velocity is maximum. The velocity can be expressed as v = ฯ‰โˆš(Aยฒ - xยฒ), where ฯ‰ is the angular frequency, A is the amplitude, and x is the displacement from the equilibrium position.

\[ v = \omega \sqrt{A^2 - x^2} \]
Calculating Kinetic Energy

The kinetic energy can be calculated using the formula KE = (1/2)mvยฒ. Substituting the expression for velocity from the previous step, we get KE = (1/2)m(ฯ‰ยฒ(Aยฒ - xยฒ)). This shows how kinetic energy varies with displacement in SHM.

\[ KE = \frac{1}{2} m \omega^2 (A^2 - x^2) \]

Key Points

  • ๐ŸŽฏ Kinetic energy in SHM is maximum at the equilibrium position.
  • ๐ŸŽฏ Kinetic energy is zero at the maximum displacement (amplitude).
  • ๐ŸŽฏ The total mechanical energy in SHM is constant and is the sum of kinetic and potential energy.

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Examples:💡

Calculate the kinetic energy of a mass of 2 kg oscillating in SHM with an amplitude of 0.5 m at a displacement of 0.3 m. Given that the angular frequency is 4 rad/s.

Solution:

Step 1: First, calculate the velocity using v = ฯ‰โˆš(Aยฒ - xยฒ). Here, A = 0.5 m, x = 0.3 m, and ฯ‰ = 4 rad/s.

\[ v = 4 \sqrt{0.5^2 - 0.3^2} \]

Step 2: Calculate the velocity: v = 4โˆš(0.25 - 0.09) = 4โˆš0.16 = 4 * 0.4 = 1.6 m/s.

\[ v = 1.6 m/s \]

Step 3: Now, use the kinetic energy formula KE = (1/2)mvยฒ. Substitute m = 2 kg and v = 1.6 m/s.

\[ KE = \frac{1}{2} \times 2 \times (1.6)^2 \]

Step 4: Calculate the kinetic energy: KE = 1 * 2.56 = 2.56 J.

Common Mistakes

  • Mistake: Confusing maximum velocity with average velocity in SHM.

    Correction: Remember that maximum velocity occurs at the equilibrium position, not the average.

  • Mistake: Forgetting to convert units when using the formulas.

    Correction: Always check that units are consistent (e.g., meters for displacement, seconds for time).

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