we know that a particle in Harmonic Motion is moving at v1 @ position x1 and v2 @ position x2.
What is A (Amplitude) and w (omega) ?
To find the amplitude (A) and angular frequency (ω) of a particle in harmonic motion, we need to know its position and velocity at two different points in time.
First, let's define some terms:
- Position (x): It refers to the displacement of the particle from its equilibrium position. In harmonic motion, the particle oscillates back and forth around the equilibrium position.
- Velocity (v): It represents the rate of change of position and indicates the particle's speed and direction of motion.
Now, let's consider the given information:
v1 @ position x1: This means that at position x1, the particle's velocity is v1.
v2 @ position x2: This means that at position x2, the particle's velocity is v2.
To calculate the amplitude (A), we can use the fact that amplitude is the maximum displacement of the particle from the equilibrium position. At these two points (x1 and x2), the particle is momentarily at rest (since it is changing direction). Thus, the displacement from the equilibrium position at these points gives us the amplitude.
A = |x2 - x1|
To find the angular frequency (ω), we can use the relationship between velocity and displacement in harmonic motion. The velocity of a particle in harmonic motion can be expressed as the derivative of position with respect to time:
v(t) = -Aωsin(ωt)
Here, t represents time.
Since the particle is momentarily at rest at positions x1 and x2, the velocities v1 and v2 are both zero. To find ω, we can set these velocities to zero and solve for ω:
0 = -Aωsin(ωt)
Since sin(ωt) cannot be zero because ωt represents the phase of harmonic motion, the only way for this equation to hold is if ω = 0.
Therefore, in the given scenario where v1 @ position x1 and v2 @ position x2, the angular frequency (ω) is zero.
To summarize:
- Amplitude (A) can be calculated using A = |x2 - x1|.
- Angular frequency (ω) is zero in this scenario.