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? $$ d(t)=4+3 \sin (\pi t) $$

Short answer

(a) The object oscillates around 4 meters with a max displacement of 3 meters. (b) 3 meters. (c) 2 seconds. (d) 0.5 Hz.

Step-by-step solution

- Identify the components of the displacement equation

The displacement equation given is: \[ d(t) = 4 + 3 \sin (\pi t) \] Here, the displacement function has two major components: 1. A constant term: 4 2. A periodic sinusoidal term: 3 \sin(\pi t)

- Describe the motion of the object

The constant term (4) shifts the entire sine curve vertically upwards by 4 units. Therefore, the object oscillates up and down around the line \(d = 4\). The amplitude of the sine function, which is 3, dictates the maximum displacement from this central line.

- Identify the maximum displacement from the rest position

The amplitude of the periodic term is 3, so the object oscillates between \(4 + 3 = 7\) meters and \(4 - 3 = 1\) meter. Hence, the maximum displacement from the rest position of 4 meters is 3 meters.

- Determine the time required for one oscillation

The period of the sine function \(d(t) = 4 + 3 \sin(\pi t)\) is the time it takes to complete one full cycle. The standard period of \(\sin(kt)\) is \(\frac{2\pi}{k}\). Here \(k = \pi\), so the period \(T = \frac{2\pi}{\pi} = 2\) seconds.

- Calculate the frequency

Frequency \(f\) is the reciprocal of the period \(T\). Thus, \[ f = \frac{1}{T} = \frac{1}{2} \] The frequency is 0.5 Hz.

Key concepts

Trigonometric Motion

Trigonometric motion refers to the movement described by trigonometric functions such as sine and cosine. In our example, the object's displacement over time is given by the function \(d(t) = 4 + 3 \, \text{sin}(\pi t)\). Here, the sine function represents the oscillatory motion typical in trigonometry.

Essentially, trigonometric motion is periodic, meaning it repeats at regular intervals. This can often be visualized as an object moving back and forth, like a pendulum. The displacement function tells us the position of the object at any given time, factoring in both a constant term and a sinusoidal term.

• Constant term (4): Shifts the entire motion vertically by 4 units.
• Sinusoidal term (3 \, \text{sin}(\pi t)): Creates the oscillations around the central line defined by the constant term.

Understanding these components helps explain the overall motion, which in this case, means the object oscillates around the line \(d = 4\).

Oscillations

Oscillations refer to the repetitive variation in displacement of an object over time. In the given displacement function \(d(t) = 4 + 3 \, \text{sin}(\pi t)\), the oscillatory motion is dictated by the sinusoidal term.

Consider an object suspended from a spring. When it moves up and down, it creates an oscillatory motion similar to our function's description. The key features of oscillations include:

• Amplitude: The maximum distance from the rest position (in this function, it's 3 meters up or down from 4 meters).
• Period: The time it takes for one complete cycle of the oscillation (calculated in the Frequency Calculation section).

In practical terms, the object moves in a smooth, wave-like pattern as it oscillates between its maximum and minimum displacement values.

Frequency Calculation

The frequency of an oscillating object is how often the motion repeats per unit time. To determine this, we first need to know the period of the motion, which is the time for one full oscillation.

For our function \(d(t) = 4 + 3 \sin(\pi t)\):

• The period of \(\sin(k t)\) is given by \(\frac{2\pi}{k}\).
• Here, \(k = \pi\), so the period \(T\) is \(\frac{2\pi}{\pi} = 2\) seconds.

Once the period is known, the frequency \(f\) can be easily found as the reciprocal of the period:

• \(f = \frac{1}{T}\)
• \(f = \frac{1}{2} = 0.5\) Hz.

Thus, the object completes one oscillation every 2 seconds, resulting in a frequency of 0.5 Hz, meaning there is one complete back-and-forth motion in 2 seconds.

Maximum Displacement

Maximum displacement refers to the farthest point an object moves from its rest position during oscillation. For our trigonometric function \(d(t) = 4 + 3 \sin(\pi t)\), we calculate this based on the amplitude of the sinusoidal term.

The formula reveals the object's motion around a central line at \(d = 4\), oscillating with an amplitude of 3 meters. Therefore, the object moves between:

• Maximum displacement: \(4 + 3 = 7\) meters
• Minimum displacement: \(4 - 3 = 1\) meter

The maximum displacement (3 meters from the rest position) represents how far the object travels from its equilibrium point, emphasizing the range of its oscillatory movement.