A simple harmonic oscillator executes motion whose amplitude is 0.20 m and it completes 60 oscillations in 2 minutes. i) Calculate its time period and angular frequency. ii) If the initial phase is 45°, write expressions for instantaneous displacement, velocity and acceleration

Question

A simple harmonic oscillator executes motion whose amplitude is 0.20 m and it completes 60 oscillations in 2 minutes. i) Calculate its time period and angular frequency. ii) If the initial phase is 45°, write expressions for instantaneous displacement, velocity and acceleration

Answer 1

simple harmonic oscillator executes motion whose amplitude is 0.20 m and it completes 60 oscillations in 2 minutes. i) Calculate its time period and angular frequency. ii) If the initial phase is 45°, write expressions for instantaneous displacement, velocity and acceleration.

Answer 2

To answer your question, let's break it down into two parts:

i) Calculating the time period and angular frequency:

1. Time period (T) is the time taken for one complete cycle of oscillation. We can calculate it using the formula: T = (Total time taken) / (Number of oscillations)
In this case, the total time taken is 2 minutes (convert it to seconds) and the number of oscillations is 60.
T = (2 minutes * 60 seconds/minute) / 60 oscillations = 2 seconds

So, the time period is 2 seconds.

2. Angular frequency (ω) represents the rate of change of angle with respect to time. It is calculated using the formula: ω = (2π) / T
In this case, T = 2 seconds, so substituting the values, we get:
ω = (2π) / 2 = π rad/s

So, the angular frequency is π rad/s.

ii) Writing expressions for instantaneous displacement, velocity, and acceleration:

1. Instantaneous displacement (x) of the simple harmonic oscillator at any time t can be given by:
x = A * cos(ω * t + φ)
where A is the amplitude of the motion (0.20 m), ω is the angular frequency (π rad/s), t is the time, and φ is the initial phase (45° = π/4 rad).
Substituting these values into the equation, we have:
x = 0.20 * cos(π * t + π/4)

2. Instantaneous velocity (v) is the derivative of displacement with respect to time:
v = dx/dt
Differentiating the expression for displacement, we have:
v = -A * ω * sin(ω * t + φ)

3. Instantaneous acceleration (a) is the derivative of velocity with respect to time:
a = dv/dt
Differentiating the expression for velocity, we have:
a = -A * ω^2 * cos(ω * t + φ)

So, the expressions for instantaneous displacement, velocity, and acceleration are:
x = 0.20 * cos(π * t + π/4)
v = -0.20 * π * sin(π * t + π/4)
a = -0.20 * π^2 * cos(π * t + π/4)

I hope this explanation helps you understand how to solve this problem! Let me know if you have any further questions.