Question

In Problems \(15-22,\) the displacement \(d\) (in meters) of an object at time \(t\) (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its rest position? (c) What is the time required for one oscillation? (d) What is the frequency? $$ d(t)=5 \sin (3 t) $$

Short answer

The motion is periodic with a maximum displacement of 5 meters, a period of \frac{2 \pi}{3}\) seconds, and a frequency of \frac{3}{2 \pi}\ Hz.

Step-by-step solution

- Describe the motion of the object

The given function is in the form of a sine function, which represents periodic oscillatory motion. Since the velocity of the object changes sinusoidally with time, it implies that the object is vibrating back and forth around its rest position.

- Find the maximum displacement

The amplitude of the sine function gives the maximum displacement from the rest position. For the function, \(d(t) = 5 \sin(3t)\), the maximum displacement (amplitude) is given by the coefficient of the sine function, which is \(5 \text{meters}\).

- Determine the time required for one oscillation

The period of the function is found via the formula \[T = \frac{2\pi}{\text{B}} \], where \(B\) is the frequency coefficient inside the sine function. For \(d(t) = 5 \sin(3t)\), \(B = 3\). Thus, \[T = \frac{2 \pi}{3} \text{ seconds}\].

- Calculate the frequency

The frequency is the reciprocal of the period. Using \[T = \frac{2 \pi}{3}\], the frequency \left( f \right) is given by \left(f = \frac{1}{T} = \frac{3}{2 \pi}\right)

Key concepts

Sinusoidal Motion

Sinusoidal motion refers to motion that can be described using sine and cosine functions. The function given in the exercise, \(d(t) = 5 \sin(3t)\), represents this type of motion. This is because sine and cosine functions exhibit a repeating, wave-like pattern.
In this particular problem, the displacement of the object varies sinusoidally with time. This means the object moves back and forth in a smooth, continuous manner. It oscillates around its rest position, moving from one extreme to another.
The use of sine and cosine functions to model such motion is common in physics, especially in scenarios involving vibrations, waves, and circular motion. These functions help describe how the displacement, velocity, and acceleration of the object change over time in a periodic manner.

Amplitude

Amplitude is a key characteristic in the study of oscillatory motion. It represents the maximum distance an object moves from its rest position during its motion. In the function provided, \(d(t) = 5 \sin(3t)\), the amplitude is given by the coefficient of the sine function, which is 5.
This means the object can move as far as 5 meters away from its rest position in either direction. Amplitude is always a positive value and it tells us about the peak of the wave in the context of trigonometric functions.

• Amplitude is crucial for understanding the energy involved in vibrations and waves. A higher amplitude generally means more energy.
• The amplitude can be directly observed in graphs of sine and cosine functions, where it determines the height of the peaks and the depth of the troughs.

Period

The period is the time it takes for one complete cycle of motion. For the function \(d(t) = 5 \sin(3t)\), the period can be calculated using the formula \[T = \frac{2 \pi}{B} \], where \(B\) is the coefficient of \(t\) inside the sine function.
In our case, \(B = 3\), so the period \[T = \frac{2 \pi}{3} \] seconds. This means it takes \[\frac{2 \pi}{3} \] seconds for the object to return to its starting position and repeat its motion.
Understanding the period of an oscillating system is important because it tells us how fast the system is oscillating. In practical scenarios, the period can help determine the timing and frequency of certain events.

• In graphs of sine and cosine functions, the period is the horizontal length of one complete cycle.
• In real-world applications, period measurement is essential in fields like engineering, acoustics, and even timekeeping.

Frequency

Frequency is another fundamental concept in oscillatory motion. It is the number of complete cycles that occur in a unit of time. The frequency is the inverse of the period.
To find the frequency \(f\), you use the formula \[f = \frac{1}{T} \]. For \(T = \frac{2 \pi}{3} \), the frequency is \[f = \frac{3}{2 \pi} \] cycles per second, or Hertz (Hz).
Frequency tells us how often the oscillation repeats in a second and is an important characteristic in many areas like sound waves, electromagnetic waves, and mechanical vibrations.

• A higher frequency means more oscillations per second, implying a faster oscillating system.
• Frequency is often used in practical applications like tuning musical instruments, designing filters in electronics, and analyzing alternating current (AC) circuits.