Question

While on a visit to Minnesota ("Land of 10,000 Lakes"), you sign up to take an excursion around one of the larger lakes. When you go to the dock where the 1500-kg boat is tied, you find that the boat is bobbing up and down in the waves, executing simple harmonic motion with amplitude 20 cm. The boat takes 3.5 s to make one complete up-and-down cycle. When the boat is at its highest point, its deck is at the same height as the stationary dock. As you watch the boat bob up and down, you (mass 60 kg) begin to feel a bit woozy, due in part to the previous night's dinner of lutefisk. As a result, you refuse to board the boat unless the level of the boat's deck is within 10 cm of the dock level. How much time do you have to board the boat comfortably during each cycle of up-and-down motion?

Short answer

You have approximately 4.668 seconds to board the boat comfortably per cycle.

Step-by-step solution

Identify Given Values

First, we need to list the values given in the problem: - Mass of the boat: 1500 kg - Amplitude of motion: 20 cm = 0.2 m (conversion from cm to m) - Period of motion (T): 3.5 s - Maximum safe distance to dock: 10 cm = 0.1 m - Highest point of the boat's motion is at the same level as the dock.

Understand Simple Harmonic Motion

Simple harmonic motion (SHM) can be described by the equation \[ x(t) = A \cos\left(\frac{2\pi}{T} t\right), \]where \( x(t) \) is the displacement of the boat at time \( t \), \( A \) is the amplitude, and \( T \) is the period of the motion.The boat is at its highest point when \( x(t) = A \) and at the same level as the dock.

Determine Boat Displacement Range

Since the maximum height of the boat from the equilibrium position is \( A = 20 \) cm, the boat travels from \( -20 \) cm to \( +20 \) cm.To board comfortably, the boat needs to be within 10 cm of the dock, which translates to \(|x| \leq 10\) cm. This corresponds to the boat being between \(-10\) cm and \(+10\) cm of displacement.

Solve for Entrance Time Frame

To find how long the boat stays within the comfortable boarding range, consider how long it stays within \(-10 \leq x(t) \leq 10\) cm.Convert these into the cosine function parameters, solving for the time intervals: \[ \frac{-10}{20} \leq \cos\left(\frac{2\pi}{3.5} t\right) \leq \frac{10}{20}. \]Solving these inequalities for time:\[ 0.5 \leq \cos\left(\frac{2\pi}{3.5} t\right) \leq 1. \]

Calculate Time Intervals

The cosine function equals 0.5 at \(\pi/3\) and 5\(\pi/3\) radians. Therefore:1. First instance, \(\frac{2\pi}{3.5}t = \pi/3 \rightarrow t = \frac{3.5}{6} \approx 0.583 \, s.\)2. Second instance, \(\frac{2\pi}{3.5}t = 5\pi/3 \rightarrow t = \frac{3.5 \, \times \, 5}{6} \approx 2.917 \, s.\)Total time between these instances during which \(\cos\left(\frac{2\pi}{3.5}t\right) \geq 0.5\) is approximately 2.917 s - 0.583 s = 2.334 s in the first half cycle.Since the motion is symmetric, the total time in both halves during which the boat can be boarded is \[ 2 \times 2.334 \, s = 4.668 \, s. \]

Conclusion

The boat is within the comfortable boarding distance for approximately 4.668 seconds per each cycle of up-and-down motion.

Key concepts

Amplitude

When discussing simple harmonic motion, the term "amplitude" is crucial to understand. Amplitude is the maximum extent of a vibration or oscillation, measured from the position of equilibrium. This means it is the furthest distance from the center or equilibrium position that the object, in this case, a boat, can reach. In our scenario, the boat has an amplitude of 20 cm, or 0.2 meters. This tells us that at its peak, the boat rises 20 cm above the equilibrium position, which is its mean level. Understanding the amplitude helps in visualizing the motion and determining how much the boat oscillates as it bobs up and down.

Period of Motion

In simple harmonic motion, the "period of motion" refers to the time it takes for the object to complete one full cycle of movement. This includes movement from its maximum to its minimum and back to the maximum position. For our boating scenario, the period is 3.5 seconds. That means it takes 3.5 seconds for the boat to move from its highest point, to the lowest, and back up to the highest. The period is a consistent measure of the cycle length regardless of the amplitude. Knowing the period is crucial when planning any task related to timing, such as boarding the boat while it's in motion.

• **Key Point:** A shorter period means a faster cycle and quicker oscillations.
• **Fun Fact:** The period of any harmonic oscillator is influenced by its physical characteristics and not the amplitude of motion.

Displacement

Displacement in the context of simple harmonic motion refers to how far the object is from its equilibrium position at any given time. It is not to be confused with distance, as it indicates direction and can be positive or negative. For example, when observing the boat, displacement will vary throughout the cycle from -20 cm to +20 cm due to its maximum amplitude.
Understanding displacement is essential to determine how safe and stable the boat is when passengers like yourself plan to board. A displacement within a comfortable range of less than 10 cm from the dock level allows for safe boarding.

Cosine Function

In mathematical representations of simple harmonic motion, the cosine function is commonly used to describe the motion's pattern over time. The equation \[ x(t) = A \cos\left(\frac{2\pi}{T} t\right) \] helps visualize the displacement over time, where:

• **x(t)** is the displacement at time **t**.
• **A** is the amplitude.
• **T** is the period of motion.

By using the cosine function in our example, we assess when the boat reaches within a safe distance for boarding. This is part of using mathematical models to predict and describe oscillating movements accurately in real-life scenarios.

Equilibrium Position

The equilibrium position is a fundamental concept in understanding simple harmonic motion. It refers to the point where the object would naturally rest if there were no net forces acting upon it. For our boat, the equilibrium position is where it rests when not affected by waves or external forces.
When the boat moves above or below this level during its oscillation, it experiences displacement. The equilibrium position is essential for calculating other parameters, like amplitude and displacement, since they are measured relative to this central point.

• **Tip for Visualization:** Imagine the boat sitting still. The spot where it's floating without movement is its equilibrium position.
• **Real-World Application:** Recognizing equilibrium helps in engineering to design structures and mechanisms that can withstand oscillation and maintain balance.