Problem 80 - Algebra and Trigonometry

Question

The displacement of a spring vibrating in damped harmonic motion is given by $$y=4 e^{-3 t} \sin 2 \pi t$$ Find the times when the spring is at its equilibrium position \((y=0)\)

Step-by-Step Solution

Answer

The spring is at equilibrium at times \(t = \frac{n}{2}\) for integer \(n\).

Step 1: Understand the Equation

The displacement of the spring is given by the function \(y = 4 e^{-3t} \sin 2 \pi t\). To find when the spring is at its equilibrium position, we need to determine when \(y = 0\).

Step 2: Set the Equation to Zero

Since we are looking for the times when the displacement \(y = 0\), set the entire function equal to zero: \(4 e^{-3t} \sin 2 \pi t = 0\).

Step 3: Analyze the Product

The equation \(4 e^{-3t} \sin 2 \pi t = 0\) can be zero if either factor is zero. However, \(e^{-3t}\) can never be zero since the exponential function is always positive for real numbers. Therefore, we focus on \(\sin 2\pi t = 0\).

Step 4: Solve the Sine Equation

The sine function is zero at integer multiples of \(\pi\), i.e., \(\sin 2\pi t = 0\) when \(2\pi t = n \pi\) where \(n\) is an integer. Simplifying to solve for \(t\), we get \(t = \frac{n}{2}\).

Step 5: State the Solution

Thus, the times at which the spring is at its equilibrium position (\(y = 0\)) are given by \(t = \frac{n}{2}\), where \(n\) is an integer.

Key Concepts

Equilibrium Position

In the study of damped harmonic motion, the concept of equilibrium position is crucial. It refers to the position where the net force acting on the vibrating system is zero. For a spring system, this means the point where the spring is neither compressed nor stretched. In mathematical terms, it's when the displacement function, which describes the position of the spring over time, equals zero.
For the given exercise, the equilibrium position is achieved when the displacement function \(y = 4 e^{-3t} \sin 2\pi t\) equals zero. Since the task is to find out when the spring reaches this position, we must solve the equation \(y = 0\). The exponential term \(e^{-3t}\) can never be zero, so it is the sine function \(\sin 2\pi t\) that must equal zero. Therefore, finding these points will determine when the spring reaches its equilibrium position.
Understanding the equilibrium position helps in visualizing and analyzing the motion of oscillating systems, not just springs, but other physical systems responding to forces.

Displacement

Displacement in the context of damped harmonic motion refers to how far a point moves from its equilibrium position as it oscillates. This is a key aspect because it tells us the distance and direction of movement at any given time.
In the exercise example, the displacement is described by the equation \(y = 4 e^{-3t} \sin 2\pi t\). Here:

• The term \(4 e^{-3t}\) represents the damping factor, which decreases the amplitude of oscillation over time.
• The sine function \(\sin 2\pi t\) gives the periodic oscillation of the system.

Combining these, the equation describes how the spring starts with a larger amplitude which gradually reduces due to damping, while continuously oscillating back and forth as governed by the sine function.
Generally, understanding displacement is essential as it provides insight into how oscillating systems behave over time, especially under the influence of damping factors that gradually curb the motion.

Sine Function

The sine function is a fundamental component of harmonic motion equations as it models the repetitive oscillatory nature of the motion. In damped harmonic motion, the sine function indicates the regular back-and-forth movement of the object.
In the given problem, the displacement equation includes the term \(\sin 2\pi t\). The sine function \(\sin\) takes values between -1 and 1 and completes a full cycle from 0 back to 0 over the interval \([0, 2\pi]\). For the spring system here:

• The frequency of oscillation is determined by the coefficient of \(t\) inside the sine function, which in this case is \(2\pi\).
• Sine becomes zero at integer multiples of \(\pi\), specifically when \(2\pi t = n\pi\), thereby dictating when the spring reaches the equilibrium position.

Therefore, in contexts like the exercise, understanding how the sine function operates helps us predict the timing and nature of oscillations within any given period.