Question

Find the amplitude, period, frequency, and velocity amplitude for the motion of a particle whose distance $s$ from the origin is the given function. $$s=2\sin 3t\cos 3t$$

Short answer

Amplitude: 1, Period: $\frac{\text{π}}{3}$, Frequency: $\frac{3}{\text{π}}$, Velocity amplitude: 6.

Step-by-step solution

Simplify the function

Use the trigonometric identity $\text{sin}(A)\text{cos}(A)=\frac{1}{2}\text{sin}(2A)$ to simplify $s=2\text{sin}(3t)\text{cos}(3t)$: $\begin{array}{rlrl}s& =2\text{sin}(3t)\text{cos}(3t)& =2\times \frac{1}{2}\text{sin}(6t)& =\text{sin}(6t)\end{array}$

Determine the amplitude

In the sine function $s=\text{sin}(6t)$, the amplitude is the coefficient of the sine function. Thus, the amplitude is $1$.

Calculate the period

The period of a sine function $\text{sin}(kt)$ is given by $\frac{2\text{π}}{k}$. For $s=\text{sin}(6t)$, $k=6$. Thus, $\text{Period}=\frac{2\text{π}}{6}=\frac{\text{π}}{3}$.

Find the frequency

Frequency $u$ is the reciprocal of the period. Thus, $u=\frac{1}{\text{Period}}=\frac{3}{\text{π}}$.

Determine the velocity amplitude

The velocity amplitude is the product of the angular frequency $\text{ω}$ and the amplitude Angular frequency, $\text{ω}=2\text{πν}=6$ Thus, the velocity amplitude is $6\times 1=6$.

Key concepts

Amplitude

Amplitude is a critical concept in harmonic motion. It describes the maximum extent of oscillation from the equilibrium position. For a sine function like $s=\sin (6t)$, the amplitude is the coefficient in front of the sine function. This coefficient indicates how far the particle moves from the origin at its peak. In the given problem, the amplitude is 1, as there is no other multiplier for the sine function. To visualize, imagine the particle moving back and forth, reaching 1 unit at its furthest points from the center.

Period

The period of a harmonic function is the time it takes for one complete cycle of motion. For the function $s=\sin (6t)$, the period is calculated using the formula $\frac{2\pi }{k}$, where $k$ is the coefficient of $t$ within the sine function. Here, $k=6$, so the period is $\frac{2\pi }{6}=\frac{\pi }{3}$. This means the particle completes one full oscillation every $\frac{\pi }{3}$ units of time.

Frequency

Frequency is directly related to the period and describes how many cycles occur per unit of time. Frequency ($u$) can be found as the reciprocal of the period. In this case, with a period of $\frac{\pi }{3}$, the frequency is $\frac{3}{\pi }$. Hence, the particle completes $\frac{3}{\pi }$ oscillations every unit of time. Higher frequency means more cycles per second, indicating faster oscillation.

Velocity Amplitude

Velocity amplitude is a measure of the maximum speed of the oscillating particle. It is calculated as the product of the angular frequency and the amplitude. The angular frequency ($\omega$) is given by $2\text{piu}$, which is also $k$. In this exercise, $\omega =6$. With an amplitude of 1, the velocity amplitude is simply $6\times 1=6$. This value shows the peak velocity at which the particle moves during its harmonic motion.

Trigonometric Identity

Trigonometric identities simplify complex equations in harmonic motion. For instance, the identity $\sin (A)\cos (A)=\frac{1}{2}\sin (2A)$ can transform the given function $s=2\sin (3t)\cos (3t)$ into $s=\sin (6t)$. Simplification helps in easily identifying key parameters like amplitude and period. Trigonometric identities are essential tools in solving and understanding harmonic motion problems due to their ability to reduce and clarify functions.