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. $$ s=5 \sin (t-\pi) $$

Short answer

Amplitude: 5, Period: 2π, Frequency: 1/(2π), Velocity Amplitude: 5

Step-by-step solution

- Identify the Form of the Function

The given function is in the form of a sine wave: \[ s = 5 \, \sin (t - \pi) \]The general form of a sine wave is:\[ s = A \, \sin (\omega t + \phi) \]where:- \(A\) is the amplitude,- \(\omega\) is the angular frequency,- \(\phi\) is the phase shift.

- Find the Amplitude

In the given function \[ s = 5 \, \sin (t - \pi) \], the coefficient of the sine function is 5. Therefore, the amplitude \(A\) is:\[ A = 5 \]

- Determine the Angular Frequency

In the function \[ s = 5 \, \sin (t - \pi) \], the angular frequency \(\omega\) is the coefficient of \(t\), which is 1. Therefore, the angular frequency \(\omega\) is:\[ \omega = 1 \]

- Calculate the Period

The period \(T\) is given by the formula:\[ T = \frac{2\pi}{\omega} \]Substitute \(\omega = 1\):\[ T = \frac{2\pi}{1} = 2\pi \]

- Calculate the Frequency

The frequency \(f\) is the reciprocal of the period:\[ f = \frac{1}{T} \]Substitute \(T = 2\pi\):\[ f = \frac{1}{2\pi} \]

- Find the Velocity Amplitude

The velocity amplitude is given by the product of the amplitude and the angular frequency:\[ v_{max} = A \cdot \omega \]Substitute \(A = 5\) and \(\omega = 1\):\[ v_{max} = 5 \cdot 1 = 5 \]

Key concepts

Amplitude

In simple harmonic motion, the amplitude is the maximum extent of the oscillation from its equilibrium position. For the function given in the exercise, \[ s = 5 \, \sin (t - \pi) \], we can see that the coefficient of the sine function is 5. This coefficient directly represents the amplitude of the wave.
Therefore, the amplitude (\(A\)) of the particle’s motion is 5 units.
The amplitude tells us how far the particle moves from its central position during its oscillation. In other words, the particle will move 5 units away from the origin at its peak.

Angular Frequency

Angular frequency, denoted by \(\omega\), describes how quickly the particle oscillates back and forth. For the given function \[ s = 5 \, \sin (t - \pi) \], we look at the coefficient of \(t\) to find the angular frequency. Here, this coefficient is 1.
Thus, the angular frequency (\(\omega\)) is 1 rad/s.
Angular frequency essentially tells us the speed of rotation per unit time. Higher angular frequencies mean more rapid oscillations.

Period

The period of simple harmonic motion is the time it takes for the particle to complete one full cycle of its motion. We calculate the period (\(T\)) using the formula: \[ T = \frac{2\pi}{\omega} \]Substituting \(\omega = 1\): \[ T = \frac{2\pi}{1} = 2\pi \text{ seconds} \]The period of 2π seconds means that every 2π seconds, the particle returns to its original position and starts repeating its motion.
The period is inversely related to the angular frequency. A higher angular frequency results in a shorter period.

Frequency

Frequency refers to the number of oscillations that occur in one second. The frequency (\(f\)) is the reciprocal of the period (\(T\)). Using the previously found period: \[ f = \frac{1}{T} \]Substitute \(T = 2\pi\): \[ f = \frac{1}{2\pi} \approx 0.159\text{ Hz} \]This indicates that the particle oscillates approximately 0.159 times per second.
Frequency measures how often the motion repeats itself in one second. It is measured in Hertz (Hz). A higher frequency means the object oscillates more frequently.

Velocity Amplitude

Velocity amplitude represents the highest speed that the particle reaches during its oscillation. It is calculated by multiplying the amplitude (\(A\)) and the angular frequency (\(\omega\)). Using the values obtained: \[ v_{max} = A \cdot \omega \] Substitute \(A = 5\) and \(\omega = 1\): \[ v_{max} = 5 \cdot 1 = 5 \text{ units/second} \]This means that the maximum velocity of the particle is 5 units per second.
Velocity amplitude helps in understanding how fast the particle will be moving at the equilibrium position (the center) during its oscillation.