Problem 138 - Algebra and Trigonometry Real Mathematics, Real People

Question

The displacement from equilibrium of an oscillating weight suspended by a spring is given by $$y(t)=\frac{1}{4} \cos 6 t$$ where \(y\) is the displacement (in feet) and \(t\) is the time (in seconds) (see figure). Find the displacement when (a) \(t=0,(b) t=\frac{1}{4},\) and \((c) t=\frac{1}{2}\)

Step-by-Step Solution

Answer

The displacement at t = 0 is \(\frac{1}{4}\) feet, at t = \(\frac{1}{4}\) seconds it is 0 feet and at t = \(\frac{1}{2}\) seconds it is \(-\frac{1}{4}\) feet.

Step 1: Evaluate the displacement at t = 0

Plug t = 0 into the displacement function \(y(t) = \frac{1}{4} \cos 6t\). We get \(y(0) = \frac{1}{4} \cos(6*0)\) which simplifies to \(y(0)=\frac{1}{4}\)

Step 2: Evaluate the displacement at t = 1/4

Next, plug t = 1/4 into the displacement function. We get \(y(\frac{1}{4}) = \frac{1}{4} \cos(6*\frac{1}{4})\). Note that the argument of the cosine function is in radians. Since cosine of \(\frac{3}{2}\pi\) is 0, we have \(y(\frac{1}{4}) = 0\).

Step 3: Evaluate the displacement at t = 1/2

Finally, plug t = 1/2 into the displacement function. We get \(y(\frac{1}{2}) = \frac{1}{4} \cos(6*\frac{1}{2})\). This simplifies to \(y(\frac{1}{2}) = \frac{1}{4}\), since the cosine of \(3\pi\) is -1.

Key Concepts

Understanding Oscillations

Oscillations describe the repetitive back-and-forth movement around a central point, which is often referred to as the equilibrium position. In our exercise, the weight suspended by a spring is an example of an object undergoing oscillation. The object moves up and down along a path, continuously changing its displacement from an equilibrium point.

In physics and mathematics, oscillations can be represented with trigonometric functions such as sine and cosine. These functions are favored due to their periodic nature, making them perfect for modeling repetitive motions. In our example, the displacement is given by the function \(y(t) = \frac{1}{4} \cos(6t)\), which uses the cosine function:

• The amplitude of the oscillation is \(\frac{1}{4}\), indicating the maximum displacement from the equilibrium position.
• The angular frequency is 6, which determines how quickly the oscillations occur over time.

Understanding the amplitude and frequency is crucial to grasping the behavior of oscillating systems. These parameters help predict how far and how fast the oscillations happen.

Exploring Displacement

In simple terms, displacement refers to how far the oscillating object is from its equilibrium position at any point in time. This is a crucial concept when analyzing oscillatory motion, as it tells us where the object is throughout its movement.

In the exercise, the displacement function \(y(t) = \frac{1}{4} \cos(6t)\) gives the object's position (in feet) at a specific time \(t\) (in seconds):

• At \(t = 0\), the object is at its starting position with a displacement of \(\frac{1}{4}\) feet.
• At \(t = \frac{1}{4}\), we observe that the displacement is 0, indicating that the object is at the equilibrium position at this time.
• Finally, at \(t = \frac{1}{2}\), the displacement returns to \(\frac{1}{4}\) feet, showing that the object has moved away from the equilibrium and returned to its maximum displacement again.

These values illustrate how displacement varies over time, highlighting the periodic aspect of oscillations. By examining these points, we can see how the object swings back and forth around its equilibrium.

Equilibrium Position and Its Importance

The equilibrium position is a key concept in the study of oscillations. It is the central point around which the oscillating motion occurs. In this exercise, it represents the position where the spring's net force on the weight is zero, and if left undisturbed, the weight would stay at this position.

Every oscillating object has a "rest point," and this is particularly important:

• During the oscillation, the object moves above and below the equilibrium point.
• The equilibrium position is where potential energy is minimal, and kinetic energy is at its maximum during the motion.

Identifying the equilibrium position can help understand the dynamics of the system, such as when the object stops accelerating and changes direction. In the function \(y(t) = \frac{1}{4} \cos(6t)\), the displacement value at the equilibrium is zero. The weight passes through this point multiple times during each full cycle of oscillation, bouncing from one maximum displacement to the other.