Problem 24 - University Physics with Modern Physics

Question

Our balance is maintained, at least in part, by the endolymph fluid in the inner ear. Spinning displaces this fluid, causing dizziness. Suppose that a skater is spinning very fast at 3.0 revolutions per second about a vertical axis through the center of his head. Take the inner ear to be approximately 7.0 cm from the axis of spin. (The distance varies from person to person.) What is the radial acceleration (in m/s\(^2\) and in \(g\)'s) of the endolymph fluid?

Step-by-Step Solution

Answer

The radial acceleration is approximately 109.82 m/s\(^2\) or 11.20 g's.

Step 1: Convert Revolutions per Second to Radians per Second

Since 1 revolution is equal to \(2\pi\) radians, we need to convert the skater's spin rate into radians per second. The skater is spinning at 3.0 revolutions per second: \(3.0 \times 2\pi = 6\pi\) radians per second.

Step 2: Determine Radial Acceleration Formula

The radial acceleration \(a_r\) for an object moving in a circle is given by \(a_r = \omega^2 r\), where \(\omega\) is the angular velocity in radians per second and \(r\) is the radius in meters.

Step 3: Substitute Values into Radial Acceleration Formula

First, convert 7.0 cm to meters: \(7.0 \text{ cm} = 0.07 \text{ m}\). Then substitute the known values into the formula: \(a_r = (6\pi)^2 \times 0.07 = 11.09 \pi^2 \) m/s\(^2\).

Step 4: Calculate Radial Acceleration in m/s²

Compute the value of \(11.09 \pi^2\) to get the radial acceleration in meters per second squared: \(a_r \approx 109.82\) m/s\(^2\).

Step 5: Convert Radial Acceleration to g's

To express the radial acceleration in terms of g's (where \(1 g = 9.81\) m/s\(^2\)), divide the radial acceleration by 9.81: \(\frac{109.82}{9.81} \approx 11.20 \text{ g's}\).

Key Concepts

Radial Acceleration

Radial acceleration is a key concept in circular motion, particularly when understanding the forces acting on a rotating body. Radial acceleration, also known as centripetal acceleration, points towards the center of the circle along which an object is moving. This means it is always directed inward, regardless of the direction of the object’s actual travel. It was calculated as 109.82 m/s² in the scenario with the skater.

The formula for radial acceleration is:

• \(a_r = \omega^2 r\)

Where:

• \(\omega\) is the angular velocity in radians per second.
• \(r\) is the radius (distance from the center of the path) in meters.

Computing it gives insights into how fast the object changes direction while moving along the curved path.

Conversion of Units

Converting units is essential in physics because it ensures all measurements align with each other, making calculations accurate. In this problem, the conversion is from revolutions per second to radians per second. Since one revolution is equivalent to \(2\pi\) radians, the skater's spin is converted by multiplying by \(2\pi\) to get \(6\pi\) radians per second. This step is crucial to use the radial acceleration formula effectively.

Moreover, the unit conversion from centimeters to meters (7.0 cm to 0.07 m) ensures our calculations are consistent with the standard units used in physics equations. Always remember to:

• Multiply revolutions by \(2\pi\) to convert to radians.
• Convert centimeters to meters by dividing by 100.

These equivalent conversions enable better communication and understanding of scientific data.

Circular Motion

Circular motion describes the movement of an object along the circumference of a circle or rotation along a circular path. This concept is visible in a spinning skater's situation. The endolymph fluid in the inner ear moves in a circle around the skater’s head, causing him to experience dizziness while spinning.

Important Characteristics of Circular Motion:

• Constant speed but changing velocity due to direction changes.
• Presence of centripetal force which maintains the circular path.
• Radial acceleration directed towards the center of the motion path.

Understanding these principles helps explicate the nature of motion not just in sports but in many real-world phenomena like the operation of engines and celestial orbits.

Angular Velocity

Angular velocity is a measure of the rate of rotation, showing how fast an object spins around a specific axis. It's a vector quantity, which includes both the magnitude and the direction (in this case, around the vertical axis). For our spinning skater, the angular velocity was calculated as \(6\pi\) radians per second.

Formulaically expressed as:

• \(\omega = \frac{\Delta \theta}{\Delta t}\)

Where:

• \(\Delta \theta\) is the angle of rotation in radians.
• \(\Delta t\) is the time over which the rotation occurs.

In practice, this helps determine the speed of rotating parts of machines or the Earth’s rotation. It’s crucial in linking circular motion to linear motion for complete physical analysis.