Problem 70 - College Physics

Question

BIO The Spinning Eel. American eels are freshwater fish with long, slender bodies that we can treat as a uniform cylinder \(1.0 \mathrm{~m}\) long and 10 \(\mathrm{cm}\) in diameter. An eel compensates for its small jaw and teeth by holding onto prey with its mouth and then rapidly spinning its body around its long axis to tear off a piece of flesh. Eels have been recorded to spin at up to \(14 \mathrm{rev} / \mathrm{s}\) when feeding in this way. Although this feeding method is energetically costly, it allows the eel to feed on larger prey than it otherwise could. A field researcher uses the slow-motion feature on her phone's camera to shoot a video of an eel spinning at its maximum rate. The camera records at 120 frames per second. Through what angle does the eel rotate from one frame to the next? A. \(1^{\circ}\) B. \(10^{\circ}\) C. \(22^{\circ}\) D. \(42^{\circ}\)

Step-by-Step Solution

Answer

The eel rotates approximately 42° between frames.

Step 1: Determine Angular Velocity

First, we need to find the angular velocity of the eel in radians per second. The eel spins at \(14 \text{ revolutions per second}\). Since one revolution equals \(2\pi \text{ radians}\), the angular velocity \(\omega\) is calculated as follows:\[\omega = 14 \times 2\pi = 28\pi \text{ radians per second}.\]

Step 2: Find Time Between Frames

Next, identify the time interval between each frame, which is determined by the camera's frame rate. The camera records at 120 frames per second, so the time between each frame \(\Delta t\) is:\[\Delta t = \frac{1}{120} \text{ seconds per frame.}\]

Step 3: Calculate Angle per Frame

Using the angular velocity and the time per frame, we calculate the angle the eel rotates between frames. The angle \(\Delta \theta\) in radians is given by:\[\Delta \theta = \omega \times \Delta t = 28\pi \times \frac{1}{120}.\]

Step 4: Convert Radians to Degrees

Convert the calculated angle from radians to degrees. Since \(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\), we find:\[\Delta \theta = \frac{28\pi}{120} \text{ radians} \times \frac{180}{\pi} \text{ degrees per radian}.\]Simplify to find the result in degrees.

Step 5: Simplify the Angle

Perform the multiplication and simplification:\[\Delta \theta = \frac{28 \times 180}{120} \]\[\Delta \theta \approx 42 \text{ degrees.}\]

Step 6: Choose the Correct Answer

Among the provided options, the calculated angle \(\Delta \theta \approx 42^{\circ}\) matches option D.

Key Concepts

Angular Velocity

Angular velocity is a measure of how fast something spins around an axis, expressed in radians per second. In the spinning eel problem, the eel reaches a speed of 14 revolutions per second.
To convert this into angular velocity, it is important to know that one full revolution around a circle corresponds to an angle of \(2\pi\) radians.
Hence, the angular velocity can be calculated as follows:

• Multiply the number of revolutions per second by \(2\pi\).

The formula is:\[ \omega = 14 \times 2\pi = 28\pi \text{ radians per second} \]Understanding angular velocity is crucial in physics to analyze the motion of rotating objects. This concept extends to various applications ranging from simple rotating fans to complex planetary rotation systems.

Radians to Degrees Conversion

When calculating motion involving angles, it is often necessary to convert between radians and degrees. The primary reason for this is that while mathematicians often prefer radians because they are a natural way to describe angle in terms of the circumference of a circle, people are typically more familiar with degrees.
The conversion factor between these two units is rooted in the fact that a full circle is \(360\) degrees and \(2\pi\) radians.
The basic conversion formulas are:

• To convert from radians to degrees: \( \theta \, (\text{in degrees}) = \theta \, (\text{in radians}) \times \frac{180}{\pi} \)
• To convert from degrees to radians: \( \theta \, (\text{in radians}) = \theta \, (\text{in degrees}) \times \frac{\pi}{180} \)

For the spinning eel, we calculated the angle in radians between each frame and then converted it to degrees to obtain a more intuitive understanding of its motion, resulting in \( \Delta \theta \approx 42^{\circ} \).
This skill of converting between radians and degrees is beneficial in both practical and theoretical aspects of studying rotational motion.

Rotational Motion

Rotational motion involves objects spinning around an axis, and understanding it is crucial for comprehending the behavior of several systems. Unlike linear motion, where we consider velocity and acceleration, rotational motion focuses on angular displacement, angular velocity, and angular acceleration. For the spinning eel, examining the eel as a uniform cylinder aids in analyzing its motion mathematically.
Key concepts to understand in rotational motion include:

• Angular Displacement: Measured in radians, indicates the change in angular position, similar to linear displacement.
• Angular Velocity: As discussed, this measures how quickly an object rotates.
• Angular Acceleration: Describes the rate of change of angular velocity over time.

Eels use rapid rotational motion to feed, illustrating how understanding these mechanics aids in biological studies as well. Many objects undergo rotational motion, from the blades of a ceiling fan to the Earth itself, making this concept a fundamental one in physics.

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. In the context of the spinning eel, kinematics helps in understanding how far and how fast it moves, which involves determining angular velocity and time between frames.
The principal equations of kinematics for rotational motion are analogous to those for linear motion, but involve angles instead of distances:

• \( \theta = \theta_0 + \omega t + \frac{1}{2}\alpha t^2 \)
• \( \omega = \omega_0 + \alpha t \)
• \( \omega^2 = \omega_0^2 + 2\alpha\theta \)

Where:

• \(\theta\) is angular position,
• \(\omega\) is angular velocity,
• \(\alpha\) is angular acceleration.

In this scenario with the eel, you primarily use these principles without considering angular acceleration because we evaluate motion between frames at a steady angular velocity.
These concepts are foundational in various fields such as biomechanics, robotic motion planning, and even space missions, making an understanding of kinematics an essential element in physics.