91Ó°ÊÓ

In simple harmonic motion (SHM), displacement refers to the distance and direction from the equilibrium position to the current position of the oscillating body. The provided equation of motion is given by:
\[ x = (6.0 \text{ m}) \, \text{cos}[(3\text{Ï€ rad/s})t + \frac{\text{Ï€}}{3} \text{ rad}] \]
Here, 6.0 meters is the amplitude, maximum distance from equilibrium. At \( t = 2.0 \text{ s} \), plug the value into the equation:
\[ x = 6.0 \, \text{cos}[6\text{Ï€} + \frac{\text{Ï€}}{3}] \]
Using periodic properties of cosine function:
\[ \text{cos}(6\text{Ï€} + \frac{\text{Ï€}}{3}) = \text{cos}(\frac{\text{Ï€}}{3}) = \frac{1}{2} \]
So, \[ x = 6.0 \, \times \, \frac{1}{2} = 3.0 \text{ m} \]
This means the displacement at 2.0 seconds is 3 meters.

Velocity in SHM is the rate of change of displacement with time. To find it, derive the position function:
\[ v = \frac{dx}{dt} = - (6.0 \text{ m})(3\text{Ï€ rad/s}) \, \text{sin}[(3\text{Ï€})(2.0) + \frac{\text{Ï€}}{3}] \]
Simplify the argument:
\[ v = - (18\text{Ï€}) \, \text{sin}(6\text{Ï€} + \frac{\text{Ï€}}{3}) \]
Knowing that:
\[ \text{sin}(6\text{π} + \frac{\text{π}}{3}) = \text{sin}(\frac{\text{π}}{3}) = \frac{\text{√3}}{2} \]
Therefore:
\[ v = - (18\text{π}) \, \times \, \frac{\text{√3}}{2} = -9\text{π}\text{√3} \text{ m/s} \]
So, the velocity at time 2.0 seconds is \( -9\text{π}\text{√3} \text{ m/s} \).

Acceleration in SHM is the rate of change of velocity or the second derivative of displacement. By differentiating the velocity again, we get:
\[ a = \frac{dv}{dt} = - (18\text{Ï€}^2) \, \text{cos}[(3\text{Ï€})(2.0) + \frac{\text{Ï€}}{3}] \]
Simplify the cosine term:
\[ a = - (18\text{Ï€}^2) \, \text{cos}(6\text{Ï€} + \frac{\text{Ï€}}{3}) \]
With:
\[ \text{cos}(6\text{Ï€} + \frac{\text{Ï€}}{3}) = \frac{1}{2} \]
Then:
\[ a = - (18\text{Ï€}^2) \, \times \, \frac{1}{2} = -9\text{Ï€}^2 \text{ m/s}^2 \]
The acceleration at t = 2.0 seconds is \( -9\text{Ï€}^2 \text{ m/s}^2 \).

In SHM, the phase gives a complete description of an oscillation at a particular time. For the given equation, the phase \( \text{φ} \) is:
\( \text{φ} = (3\text{π})(2.0) + \frac{\text{π}}{3} \). Simplify:
\( \text{φ} = 6\text{π} + \frac{\text{π}}{3} \)
Phase is crucial because it combines the angular frequency and initial phase, dictating the exact position and direction of motion at any time.

Frequency \( \text{f} \) in SHM is the number of oscillations per unit time. Given the angular frequency \( \text{ω} = 3\text{π rad/s} \):
\[ f = \frac{\text{ω}}{2\text{π}} = \frac{3\text{π}}{2\text{π}} = 1.5 \text{ Hz} \].
Hence, the frequency of motion is 1.5 Hertz, meaning the body completes 1.5 oscillations every second.

The period \( \text{T} \) is the time taken for one complete cycle of motion. It is the reciprocal of frequency:
\[ T = \frac{1}{\text{f}} = \frac{1}{1.5} = \frac{2}{3} \text{ s} \].
Thus, the period of oscillation is \( \frac{2}{3} \) seconds or approximately 0.67 seconds.
This means each full oscillation takes a little less than a second to complete.