1. Understand the Equation
* x: Displacement of the particle from its equilibrium position.
* a: Amplitude of the oscillation (maximum displacement).
* ω: Angular frequency (2 in this case).
* t: Time.
2. Find the Acceleration Equation
The acceleration in simple harmonic motion is given by:
* a(t) = -ω²x(t)
* This means acceleration is proportional to the negative of the displacement.
Substitute the given equation for x(t):
* a(t) = -ω² * a cos(2t)
3. Determine the Minimum Acceleration
* Maximum of Cosine: The cosine function oscillates between -1 and 1. Its maximum value is 1.
* Minimum Acceleration: The minimum acceleration occurs when the cosine function is at its maximum value (1).
Therefore, the minimum acceleration is:
* a_min = -ω²a * 1 = -ω²a
4. Substitute the Value of ω
In this case, ω = 2, so the minimum acceleration is:
* a_min = -(2)²a = -4a
Conclusion
The minimum acceleration of the particle in simple harmonic motion described by x = a cos(2t) is -4a. The negative sign indicates that the acceleration is in the opposite direction of the displacement when the displacement is maximum.
