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. $$s=\frac{1}{2} \cos (\pi t-8)$$

Short answer

Amplitude: \( \frac{1}{2} \). Period: 2. Frequency: \( \frac{1}{2} \). Velocity amplitude: \( \frac{\pi}{2} \).

Step-by-step solution

Identify the given function

The displacement function is given as \( s = \frac{1}{2} \cos (\pi t - 8) \). This is in the standard form of a cosine function \( s = A \cos(\omega t + \phi) \).

Find the amplitude

The amplitude \(A\) is the coefficient in front of the cosine function. Here, it is \( \frac{1}{2} \).

Determine the angular frequency

The angular frequency \( \omega \) is the coefficient of \( t \) inside the cosine function. In this case, \( \omega = \pi \).

Calculate the period

The period \( T \) is given by the formula \( T = \frac{2\pi}{\omega} \). Using \( \omega = \pi \), we have \( T = \frac{2\pi}{\pi} = 2 \).

Find the frequency

The frequency \( f \) is the reciprocal of the period: \( f = \frac{1}{T} \). Using \( T = 2 \), we get \( f = \frac{1}{2} \).

Solve for velocity amplitude

The velocity amplitude is given by \( A\omega \). With \( A = \frac{1}{2} \) and \( \omega = \pi \), the velocity amplitude is \( \frac{1}{2} \cdot \pi = \frac{\pi}{2} \).

Key concepts

Amplitude

In harmonic motion, the amplitude represents the maximum displacement from the equilibrium position. It's basically how far the object moves away from the center point in its motion.
The given displacement function is: -The function can be written in the standard form of a sinusoidal function: oIn our specific problem, the equation is given as: -The amplitude here is the coefficient of the cosine function, which is .
This means the particle can move 0.5 units away from its origin or center point.

Frequency

Frequency in harmonic motion tells us how many cycles (or complete oscillations) happen in one second. Mathematically, it is the reciprocal of the period.
Given the period from our problem, which we calculated to be 2 seconds, the frequency ( f ) can be found using the relationship .
Substituting the period (T), we get:
This means the particle completes half a cycle every second. Frequency is often measured in Hertz (Hz), so here, the frequency is 0.5 Hz.

Angular Frequency

Angular frequency ( [ ) is a measure of how quickly the object goes through its cycles, expressed in radians per second. It is related to the regular frequency but takes into account the fact that one full cycle is radians.In our problem, we identified the angular frequency . This value directly came from the coefficient of t in the cosine function ( ( inside the cosine function.It tells us how many radians the particle goes through each second. Here, means the particle oscillates radians per second.
Angular frequency is crucial in understanding how quickly the particle’s position changes over time.

Period

The period (T) is the amount of time it takes for the particle to complete one full cycle of its motion.
It is closely related to angular frequency and can be calculated using the formula:

In our case, with , substituting in gives: (seconds).This tells us that it takes 2 seconds for the particle to complete one cycle of its motion, moving from its starting point back to the same point.Understanding the period helps in predicting the future positions of the particle in its motion.

Velocity Amplitude

Velocity amplitude refers to the maximum velocity the particle can achieve during its motion. It’s calculated as the product of amplitude (A) and angular frequency ( .
In the given problem, we have: which means the particle can reach a maximum velocity of This value indicates the highest speed, in units per second, that the particle can obtain during its harmonic motion. Understanding velocity amplitude helps reveal how fast the particle’s motion can get at its peak.