Question

Displacement of a simple harmonic oscillatoris expressed by the following equation.

Y = 10-2sin (314t + π/4)
Where all the quantities art taken is SI units. Find the following characteristics of its
oscillations. (i) Amplitude, (ii) Frequency, (iii) Initial Phase, (iv) Amplitude of
velocity

Answer 1

The amplitude of the simple harmonic oscillator is 2, the frequency is 314/(2π) Hz, the initial phase is π/4, and the amplitude of velocity is 0.27 x 314/(2π) x 2 m/s.

Step-by-step explanation:

The given equation represents the displacement of a simple harmonic oscillator:

Y = 10-2sin (314t + π/4)

1. Amplitude: The amplitude of the oscillator is the maximum displacement from its equilibrium position. In this equation, the amplitude is 2.
2. Frequency: The frequency of the oscillator is the number of complete oscillations it makes in one second. In this equation, the frequency is given by the coefficient of 't' which is 314/(2π) Hz.
3. Initial Phase: The initial phase is given by the constant term in the equation. In this case, it is π/4.
4. Amplitude of Velocity: The amplitude of velocity can be calculated using the equation 0.27 x frequency x amplitude. In this case, the amplitude of velocity is 0.27 x 314/(2π) x 2 m/s.

Answer 2

The amplitude of oscillation is 2 m, frequency is approximately 50 Hz, initial phase is π/4, and amplitude of velocity is 628 m/s for the given simple harmonic oscillator.

The displacement of a simple harmonic oscillator is given by the equation Y = 10 - 2sin(314t + π/4).

(i) Amplitude (A): The amplitude is the coefficient in front of the sine function, which is the maximum displacement the oscillator can have from its equilibrium position. In this equation, the amplitude would be 2.

(ii) Frequency (f): The frequency is found by dividing the angular frequency by 2π. The angular frequency (ω) here is 314 rad/s, so the frequency is ω/2π, which is 314/2π Hz.

(iii) Initial Phase (φ): The initial phase is the phase constant in the sine function. It's given as π/4 in this equation.

(iv) Amplitude of velocity (Umax): The amplitude of velocity is given by the product of the angular frequency (ω) and the amplitude (A). Umax = ω * A, which would be 314 * 2 m/s in this case.

To summarize, the amplitude of this simple harmonic oscillator is 2 m, the frequency is 314/2π Hz, the initial phase is π/4, and the amplitude of velocity is 628 m/s.