Problem 1 - Fundamentals of Physics

Question

If a wave \(y(x, t)=(6.0 \mathrm{~mm}) \sin (k x+(600 \mathrm{rad} / \mathrm{s}) t+\phi)\) travels along a string, how much time does any given point on the string take to move between displacements \(y=+2.0 \mathrm{~mm}\) and \(y=-2.0 \mathrm{~mm}\) ?

Step-by-Step Solution

Answer

The time is approximately 5.24 milliseconds.

Step 1: Identify the Given Wave Equation

The wave equation is given as \( y(x, t) = (6.0 \text{ mm}) \sin(k x + 600 \text{ rad/s} \cdot t + \phi) \). Here, the amplitude \( A = 6.0 \text{ mm} \), angular frequency \( \omega = 600 \text{ rad/s} \). We are tasked with finding the time it takes for the displacement to move from \( y = +2.0 \text{ mm} \) to \( y = -2.0 \text{ mm} \).

Step 2: Find the Phase Corresponding to Displacements

For the displacement \( y = 2.0 \text{ mm} \), solve for the phase \( \theta_1 \) in terms of \( t \): \( 2.0 \text{ mm} = 6.0 \text{ mm} \cdot \sin(\theta_1) \). This simplifies to \( \sin(\theta_1) = \frac{1}{3} \). Similarly, for \( y = -2.0 \text{ mm} \), solve for the phase \( \theta_2 \): \( -2.0 \text{ mm} = 6.0 \text{ mm} \cdot \sin(\theta_2) \) leading to \( \sin(\theta_2) = -\frac{1}{3} \).

Step 3: Determine Phase Angles

For \( \sin(\theta_1) = \frac{1}{3} \), we have \( \theta_1 = \arcsin\left(\frac{1}{3}\right) \) and \( \pi - \arcsin\left(\frac{1}{3}\right) \). For \( \sin(\theta_2) = -\frac{1}{3} \), \( \theta_2 = -\arcsin\left(\frac{1}{3}\right) \) and \( \pi + \arcsin\left(\frac{1}{3}\right) \). Choose consecutive angles for the wave moving smoothly from \( +2.0 \text{ mm} \) to \( -2.0 \text{ mm} \).

Step 4: Select Consecutive Phase Angles

Choose consecutive phase angles \( \theta_1 = \arcsin\left(\frac{1}{3}\right) \) and \( \theta_2 = \pi + \arcsin\left(\frac{1}{3}\right) \) that correspond to the required transition from \( +2.0 \text{ mm} \) to \( -2.0 \text{ mm} \).

Step 5: Calculate the Time Interval

Since \( \omega = 600 \text{ rad/s} \), the time difference \( \Delta t \) between these angles is \( \Delta t = \frac{\theta_2 - \theta_1}{\omega} = \frac{\left( \pi + \arcsin\left(\frac{1}{3}\right) \right) - \arcsin\left(\frac{1}{3}\right)}{600} \). Simplifying gives \( \Delta t = \frac{\pi}{600} \). Evaluate \( \Delta t \) using \( \pi \approx 3.1416 \).

Step 6: Solve for Numeric Value of Time Interval

Compute \( \Delta t = \frac{3.1416}{600} \approx 0.00524 \text{ seconds} \) or \( 5.24 \text{ milliseconds} \).

Key Concepts

Amplitude

In the context of wave equations, **Amplitude** is a crucial concept. Simply put, it is the maximum displacement from the wave's equilibrium position. In the given wave equation, the amplitude is expressed as 6.0 millimeters. This means that the maximum height (or depth) that the wave can reach from its central axis is 6.0 millimeters.

• Amplitude affects the energy of the wave—higher amplitudes indicate more energy.
• Mathematically, it is the coefficient in front of the sine function in the wave equation.

Understanding amplitude helps in visualizing how "tall" a wave can extend as it oscillates, resembling peaks and valleys.

Angular Frequency

**Angular Frequency** relates to how quickly the wave oscillates over time. For our wave, the angular frequency is 600 rad/s.

• Angled frequency is denoted by the Greek letter \( \omega \).
• It contributes to the calculation of the wave's periodic motion.

Angular frequency is critical as it gives insight into how rapid the wave oscillates. A higher angular frequency signifies that the wave completes more cycles in a given time frame.

Phase Angle

The **Phase Angle** can be a bit tricky but is essential for understanding where exactly the wave is at any given moment in its cycle. This angle shifts the wave forward or backward in time without altering its form.

• Phase angle changes the starting point of the wave.
• In our problem, it is represented as \( \phi \) in the wave equation.

Phase angles are particularly important when comparing multiple waves. They tell us how "in-step" (or "out-of-step") these waves might be relative to one another.

Sinusoidal Wave

A **Sinusoidal Wave** is a wave that describes a smooth periodic oscillation. It's one of the fundamental types of wave shapes and is given by the sine function \( \sin(\cdot) \).

• The wave equation we discuss is a sinusoidal wave equation.
• Such waves are important because they model various natural phenomena like sound and light.

A sinusoidal wave is pleasingly symmetrical, characterized by its continuous wave of peaks and troughs. Mastering this concept eases the study of more complex wave behaviors.

Time Interval Calculation

**Time Interval Calculation** is pivotal when determining how long it takes for particular events to occur—such as reaching specific points on a wave.

In the provided exercise, we calculate the time it takes for a point to go from a displacement of +2.0 mm to -2.0 mm:

• Understand the change in phase needed for this movement.
• Apply the formula \( \Delta t = \frac{\Delta \theta}{\omega} \) where \( \Delta \theta \) is the change in phase.
• Substitute angular frequency and phase angles into the formula.

In the example, we calculated the time interval as approximately 5.24 milliseconds, demonstrating the change in vertical position on the wave within a concise duration.