Chapter 7: Problem 2

Find the amplitude, period, frequency, and velocity amplitude for the motion of a particle whose distance $s$ from the origin is the given function. $$s=2 \sin (4 t-1)$$

Short Answer

Expert verified

Amplitude = 2, Period = $ \frac{\pi}{2} $, Frequency = $ \frac{2}{\pi} $, Velocity Amplitude = 8

Step by step solution

Identify the form of the sine function

The given function is of the form \[ s = A \, \sin(Bt + C) \]By comparing, we find:\[ A = 2 \]\[ B = 4 \]\[ C = -1 \]

Find the amplitude

Amplitude is the coefficient of the sine function, which is: \[ A = 2 \]

Calculate the period

The period of a sine function is given by:\[ \text{Period} = \frac{2 \pi}{B} \]Substituting the value of $ B $: \[ \text{Period} = \frac{2 \pi}{4} = \frac{\pi}{2} \]

Determine the frequency

Frequency is the reciprocal of the period:\[ \text{Frequency} = \frac{1}{\text{Period}} = \frac{1}{\pi/2} = \frac{2}{\pi} \]

Calculate the velocity amplitude

Velocity is the derivative of displacement with respect to time. To find this, first find the derivative of \[ s = 2 \, \sin(4t - 1) \]: \[ v = \frac{ds}{dt} = 2 \, \cdot 4 \, \cos(4t - 1) = 8 \cos(4t - 1) \]The amplitude of velocity is the coefficient of the cosine function:\[ \text{Velocity Amplitude} = 8 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

amplitude

In harmonic motion, amplitude refers to the maximum distance a particle moves from its equilibrium position. In the given function, the amplitude is represented by the coefficient of the sine function. For the function: $$s = 2 \, \sin(4t - 1)$$ The amplitude is 2. This tells us that the particle moves 2 units away from the origin at its maximum displacement.

period

The period of harmonic motion is the time it takes for the particle to complete one full cycle of motion. For a sine function like $$s = A \, \sin(Bt + C)$$ the period is calculated by the formula: $$\text{Period} = \frac{2 \pi}{B}$$ By substituting the value of B from our function: $$\text{Period} = \frac{2 \pi}{4} = \frac{\pi}{2}$$ Thus, the period of the particle's motion is $\frac{\pi}{2}$ seconds. This means it takes $\frac{\pi}{2}$ seconds to complete one full cycle from its starting point and back.

frequency

Frequency tells us how many cycles of motion occur per unit of time. It is the reciprocal of the period. Using our periodic function: $s = 2 \sin(4t - 1)$, the period was found to be $\frac{\pi}{2}$ seconds. Using the formula: \[\text{Frequency} = \frac{1}{\text{Period}}\] we find: \[\text{Frequency} = \frac{1}{(\pi / 2)} = \frac{2}{\pi}\] Therefore, the frequency of the motion is $\frac{2}{\pi}$ cycles per second, indicating approximately how many cycles the particle completes each second.

velocity amplitude

Velocity amplitude is related to the maximum speed the particle reaches during its motion. To find this, we consider the time derivative of the displacement function. For $$s = 2 \sin(4t - 1)$$, the velocity is: $$v = \frac{\mathrm{d}s}{\mathrm{dt}} = 2 \cdot 4 \cos(4t - 1) = 8 \cos(4t - 1)$$. The amplitude of this velocity, or the maximum speed, is the coefficient of the $\cos(4t - 1)$ term, which is 8. So, the velocity amplitude is 8 units per second, indicating the particle reaches this speed at its fastest point in the cycle.

Find the amplitude, period, frequency, and velocity amplitude for the motion of a particle whose distance \(s\) from the origin is the given function. $$s=2 \sin (4 t-1)$$