Problem 19 - Algebra and Trigonometry

Question

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? $$ d(t)=-9 \sin \left(\frac{1}{4} t\right) $$

Step-by-Step Solution

Answer

The motion is simple harmonic. The maximum displacement is 9 meters. The period is 8\pi seconds and the frequency is 1/(8\pi) Hz.

Step 1: Understand the given function

The displacement function is given by \[ d(t) = -9 \sin \left( \frac{1}{4} t \right) \].This function describes a sinusoidal motion, indicating it is periodic.

Step 2: Describe the motion of the object

Since the function is a sine function, the motion of the object is simple harmonic motion. The negative sign indicates that the motion starts in the opposite direction of the positive sine function.

Step 3: Find the maximum displacement

The maximum value of \( \sin(x) \) is 1. Thus,\[ d(t) = -9 \cdot 1 = -9 \text{ meters} \].The maximum displacement from the rest position is 9 meters.

Step 4: Calculate the time for one oscillation (period)

The period \(T\) of a sine function \( A \sin(Bt) \) is given by \( T = \frac{2\pi}{B} \). Here, \( B = \frac{1}{4} \). Thus,\[ T = \frac{2\pi}{\frac{1}{4}} = 8\pi \text{ seconds} \].

Step 5: Determine the frequency

Frequency \( f \) is the reciprocal of the period. Thus,\[ f = \frac{1}{T} = \frac{1}{8\pi} \text{ Hz} \].

Key Concepts

sinusoidal function

A sinusoidal function is a mathematical function that describes a smooth and repetitive oscillation. It is often depicted as a sine or cosine wave.

If we look at the given function, \(d(t) = -9 \sin \left( \frac{1}{4} t \right)\), it directly reflects sinusoidal behaviour. Here, \(\sin(x)\) represents the sine function, which oscillates between -1 and 1 as its argument changes.

The negative sign in front of the sine function simply means that the wave is inverted or flipped upside down. This means that the motion starts in the opposite direction of a standard positive sine wave.

Sinusoidal functions are incredibly useful because they model periodic phenomena, such as sound waves, light waves, and even the motion of a pendulum or spring.

maximum displacement

Maximum displacement refers to the furthest point an object moves from its starting or rest position during oscillation.

In the given function \(d(t) = -9 \sin \left( \frac{1}{4} t \right)\), the maximum value \(\sin(x)\) can take is 1. Hence, the maximum displacement from the rest position is calculated as follows:
\( d(t) = -9 \cdot 1 = -9 \text{ meters}\).

Absolute displacement removes the negative sign to give an absolute value, meaning the object moves 9 meters away from the rest position at its peak. Maximum displacement is an important concept because it tells us the amplitude of the motion, indicating the energy involved in the movement.

oscillation period

The oscillation period is the time it takes for an object to complete one full cycle of motion from its starting point, back to that point.

For the function \(d(t) = -9 \sin \left( \frac{1}{4} t \right)\), the period \(T\) is given by the formula \( T = \frac{2\pi}{B} \).

Here \( B = \frac{1}{4} \), substituting this in gives:
\( T = \frac{2\pi}{\frac{1}{4}} = 8\pi \text{ seconds}\).

This means the object takes \(8\pi\) seconds to complete one oscillation.

Knowing the period is crucial because it helps in understanding the timing and rhythm of the system's movements, and is a fundamental property of any periodic motion.

frequency calculation

Frequency describes how often an oscillation repeats in one second. It is the reciprocal of the period and is measured in Hertz (Hz).

For the given function, knowing the period \(T = 8\pi\) seconds, we find the frequency using the relation:
\( f = \frac{1}{T} \text{ Hz}\).

So, our calculation is:
\( f = \frac{1}{8pi} \text{ Hz}\).

This very small frequency value indicates that each oscillation takes a substantial amount of time to complete. Understanding frequency is vital in many real-world applications, such as tuning musical instruments, designing circuits, and studying wave phenomena.