Problem 89 - University Physics with Modern Physics

Question

Don't Miss the Boat. 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 \(1500-\mathrm{kg}\) boat is tied, you find that the boat is bobbing up and down in the waves, executing simple harmonic motion with amplitude 20 \(\mathrm{cm} .\) The boat takes 3.5 \(\mathrm{s}\) 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 \(\mathrm{kg}\) ) 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 \(\mathrm{cm}\) of the dock level. How much time do you have to board the boat comfortably during each cycle of up-and-down motion?

Step-by-Step Solution

Answer

You have about 1.17 seconds to board the boat during each cycle.

Step 1: Determine the equation of motion

The motion of the boat can be described by the equation for simple harmonic motion. We have amplitude \(A = 0.20\, \text{m}\) and period \(T = 3.5\, \text{s}\). The equation for vertical displacement \(y\) as a function of time \(t\) is given by: \[ y(t) = A \cos\left( \frac{2\pi}{T} t \right) \]where \(y(t)\) is the vertical displacement from the equilibrium position.

Step 2: Calculate vertical displacement limits

We need the deck to be within \(10\, \text{cm}\) (or \(0.10\, \text{m}\)) of the dock level to safely board. Since the dock and equilibrium point of the boat occur when \(y = 0\), the deck should stay between \(-0.10\, \text{m}\) and \(+0.10\, \text{m}\).

Step 3: Solve for time intervals during which the deck height is appropriate

We solve for \(t\) when \(-0.10 \leq 0.20 \cos\left(\frac{2\pi}{3.5} t\right) \leq 0.10\). This simplifies to finding \(t\) when\[\cos\left(\frac{2\pi}{3.5} t\right) \leq \frac{1}{2}\]and\[\cos\left(\frac{2\pi}{3.5} t\right) \geq -\frac{1}{2}\].Using cosine properties, these inequalities tell us the deck is within the required range for one-third of the cycle (equivalent to \(120^\circ\) or \(240^\circ\) intervals centered at equilibrium).

Step 4: Calculate the time within appropriate condition

The period \(T = 3.5\, \text{s}\), so one full cycle takes that amount of time. Since we want one-third of the cycle:\[\text{Time Comfortable} = \frac{1}{3} \times 3.5 \approx 1.17\, \text{s}.\]This means you can board the boat safely for about \(1.17\, \text{s}\) per cycle.

Key Concepts

Equation of Motion

In simple harmonic motion, the equation of motion is essential to describing how an object moves over time. For the boat bobbing up and down in the water, we use the equation for vertical displacement which is given by:\[y(t) = A \cos\left( \frac{2\pi}{T} t \right)\]where:

• \(y(t)\) is the vertical displacement from the equilibrium position.
• \(A\) is the amplitude, representing the maximum displacement, in this case 20 cm or 0.20 m.
• \(T\) is the period, the time taken for one complete oscillation, here 3.5 s.
• \(t\) is the time in seconds.

This equation indicates that the vertical displacement of the boat follows a cosine wave pattern, oscillating between the values of \(\pm A\). The cosine function is particularly suited for this kind of motion as it naturally describes repetitive up and down movement—such as the bobbing boat.

Vertical Displacement

Vertical displacement in simple harmonic motion concerns how the position of the object changes over time relative to its equilibrium. In the case of the boat, the equilibrium position is when the dock and the deck are level with each other. Due to the amplitude, the boat's deck can move up and down by as much as 20 cm, meaning the vertical displacement varies between -0.20 m and +0.20 m. However, for boarding, it’s important to find when the deck is within 10 cm of equilibrium. This creates boundaries: the deck should stay between -0.10 m and +0.10 m for safe boarding. Mathematically, this is expressed as:\[-0.10 \leq y(t) \leq +0.10\]When solving these inequalities, we look for the times when the deck height meets this criterion so the deck isn’t too high or too low, which is critical when timing your boarding.

Period of Oscillation

The period of oscillation is a fundamental property in simple harmonic motion, representing how long it takes for one complete cycle of motion. For the boat exercise, this period \(T\) is given as 3.5 seconds.Knowing the period allows us to understand not just how often the boat bobs up and down, but also helps determine when it's safe to board. If the period is 3.5 seconds, during each cycle there is a limited time frame when the boat's deck is close enough to the dock level (within the 10 cm range).By solving the inequalities, we find that the deck remains within this safe zone for about one-third of the cycle. Thus, you can board the boat comfortably for approximately 1.17 seconds per cycle. Understanding the period helps us anticipate safe boarding intervals, crucial for passengers sensitive to movements or the uncertain nature of boat decks.