Problem 37 - College Physics

Question

Dizziness. Our balance is maintained, at least in part, by the endolymph fluid in the inner ear. Spinning displaces this fluid, causing dizziness. Suppose a dancer (or skater) is spinning at a very high 3. 0 revolutions per second about a vertical axis through the center of his head. Although the distance varies from person to person, the inner ear is approximately 7.0 \(\mathrm{cm}\) from the axis of spin. What is the radial acceleration (in \(\mathrm{m} / \mathrm{s}^{2}\) and in \(g^{\prime} s\) s of the endolymph fluid?

Step-by-Step Solution

Answer

The radial acceleration is 44.46 \(\text{m/s}^2\) or approximately 4.54 \(g's\).

Step 1: Convert Revolutions per Second to Radians per Second

Given that the dancer is spinning at 3.0 revolutions per second, we need to convert this angular velocity into radians per second. Since one revolution is equal to \(2\pi\) radians, the angular velocity \(\omega\) in radians per second is:\[\omega = 3.0 \times 2\pi = 6\pi \text{ rad/s}\]

Step 2: Calculate the Radial Acceleration

Radial (centripetal) acceleration \(a_c\) is given by the formula:\[a_c = \omega^2 \times r\]where \(r\) is the radius of the circle, which is the distance from the axis of rotation to the inner ear, 7.0 cm or 0.07 m. Plugging in the values, we have:\[a_c = (6\pi)^2 \times 0.07 = 44.46 \text{ m/s}^2\]

Step 3: Convert Radial Acceleration to g's

The acceleration due to gravity \(g\) is \(9.8 \text{ m/s}^2\). To express the radial acceleration in terms of \(g\)'s, divide the radial acceleration by \(g\):\[\frac{a_c}{g} = \frac{44.46}{9.8} \approx 4.54 \, g's\]

Key Concepts

Centripetal Acceleration

Centripetal acceleration is an essential concept when dealing with objects in circular motion. It refers to the acceleration directed towards the center of the circle that keeps the object moving along a curved path. In simpler terms, it's what prevents objects from flying off in a straight line when spinning around a center.
Imagine swinging a ball on a string; the string pulls the ball into the center of the circle, keeping it on its path. This inward acceleration is necessary to change the direction of the velocity vector of the object, even if its speed remains constant. The magnitude of centripetal acceleration \( a_c \) is calculated using the formula:
\[ a_c = \omega^2 \times r \]
where \( \omega \) is the angular velocity and \( r \) is the radius of the circle. Centripetal acceleration is crucial in various applications, from vehicle dynamics in curves to understanding how the Earth orbits the Sun.

Angular Velocity

Angular velocity refers to the rate at which an object rotates around a particular axis. It's a measure of how quickly something spins and is expressed in radians per second. In the case of a dancer or skater, angular velocity tells us how fast they spin around their vertical axis.

• The formula for angular velocity \( \omega \) is:

\[ \omega = \text{number of revolutions per second} \times 2\pi \]
In the given exercise, the dancer spins at 3 revolutions per second. Converting this into radians involves multiplying by \( 2\pi \), which gives an angular velocity of
\( 6\pi \) radians per second. Angular velocity is linked with concepts like rotational kinetic energy and plays a significant role in disciplines such as engineering, astronomy, and biomechanics.

Endolymph Fluid

Endolymph fluid is a key player in our sense of balance, nestled within the labyrinth of our inner ears. This fluid fills the semicircular canals, which detect changes in head rotation. When you spin or turn your head, the endolymph moves, helping the brain to decipher movement.
When a dancer spins, the endolymph fluid within the ear's canals gets displaced due to the centripetal forces acting on it. This movement sends signals to the brain, creating the sense of rotation. However, if one spins rapidly, the fluid continues moving even after the spinning stops, contributing to the feeling of dizziness and disorientation often experienced.
The role of endolymph fluid goes beyond just balance; it also assists in stabilizing vision during rapid head movement, which is vital for performers and athletes who require precision in fast-paced routines.

Balance in Inner Ear

Balance in the inner ear is a fascinating and complex mechanism largely controlled by the vestibular system. This system consists of three main components—the semicircular canals, otolithic organs, and vestibular nerve—all filled or influenced by endolymph fluid.

• The semicircular canals respond to rotational movements, thanks to their orientation in the three planes of space.
• The otolithic organs detect linear accelerations and the position of the head relative to gravity.

Together, these detect and respond to movement, providing the brain with essential information to keep us upright and oriented. The information processed by the inner ear is critical not only for maintaining balance but also for coordinating movements and gaze stabilization.
When the inner ear's balance system detects a change, like rapid spinning, the feedback can cause an imbalance in signals, leading to dizziness or vertigo. This demonstrates the inner ear's pivotal role in our everyday movements, enabling us to perform tasks seamlessly.