Problem 28 - College Physics

Question

\(\bullet\) Dental hygiene. Electric toothbrushes can be effective in removing dental plaque. One model consists of a head 1.1 cm in diameter that rotates back and forth through a \(70.0^{\circ}\) angle 7600 times per minute. The rim of the head contains a thin row of bristles. (See Figure 9.25.) (a) What is the average angular speed in each direction of the rotating head, in rad/s? (b) What is the average linear speed in each direction of the bristles against the teeth? (c) Using your own observations, what is the approximate speed of the bristles against your teeth when you brush by hand with an ordinary toothbrush?

Step-by-Step Solution

Answer

(a) 98.89 rad/s, (b) 0.544 m/s, (c) around 0.1 m/s.

Step 1: Calculate Angular Displacement

We are given that the head rotates through a \(70.0^{\circ}\) angle. First, convert degrees to radians using the conversion factor \( \pi \, \text{radians} = 180^{\circ} \). Thus,\[ 70.0^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{7\pi}{18} \, \text{radians} \approx 1.222 \, \text{radians} \]

Step 2: Determine Full Cycle Angular Displacement

Since the head rotates through a \(70.0^{\circ}\) angle and then back, the total angular displacement for one cycle is twice the angle (going forward and back). Therefore:\[ 2 \times \frac{7\pi}{18} = \frac{7\pi}{9} \approx 2.444 \, \text{radians} \]

Step 3: Find Average Angular Speed

Calculate the angular speed per one full cycle.- The toothbrush performs 7600 cycles per minute.- Therefore, it completes \( \frac{7600}{60} \approx 126.67 \) cycles per second.- Hence, the average angular speed \( \omega \) is:\[ \omega = 126.67 \, \text{cycles/s} \times \frac{7\pi}{9} \, \text{radians/cycle} \approx 98.89 \, \text{rad/s} \]

Step 4: Calculate Average Linear Speed

The average linear speed \( v \) relates to angular speed \( \omega \) by the formula \( v = r \cdot \omega \), where \( r \) is the radius. Given the head diameter is \(1.1 \, \text{cm} \), its radius \( r \) is half:\[ r = \frac{1.1 \, \text{cm}}{2} = 0.55 \, \text{cm} = 0.0055 \, \text{m} \]Thus, the linear speed is:\[ v = 0.0055 \, \text{m} \times 98.89 \, \text{rad/s} \approx 0.544 \, \text{m/s} \]

Step 5: Estimate Hand Brushing Speed

Observationally, estimate the speed of hand brushing. Typically, one may move a toothbrush about 30 times per minute with a stroke length of approximately 5 cm. Hence, each brushing stroke takes approximately 2 seconds.Speed for one stroke:\[ v = \frac{5 \, \text{cm}}{2 \, \text{s}} \approx 0.025 \, \text{m/s} \]Typically, manual brushing would not exceed \(0.1 \, \text{m/s}\) speed.

Key Concepts

Linear Speed Calculation

Understanding linear speed is crucial, especially when it translates angular motion into movement along a line. In the world of electric toothbrushes, this concept is applied when converting the rotational motion of the brush head into the speed at which the bristles move across your teeth. This smooth linear motion is what directly interacts with dental surfaces to clean plaque.
To calculate linear speed (\( v \)), we use the formula:

• \( v = r \cdot \omega \)

Here, \( r \) is the radius of the rotating brush head, and \( \omega \) is the angular speed. In the exercise, we found the angular speed to be approximately 98.89 \( \text{rad/s} \) and the radius to be 0.0055 meters. Substituting these values gives the linear speed of the bristles as approximately 0.544 \( \text{m/s} \).
Linear speed helps us understand how effectively the toothbrush removes plaque by gauging how fast the bristles move against the teeth surface. This concept is not only applicable to electric toothbrushes but also to various mechanical systems involving rotating parts.

Electric Toothbrush Mechanics

The mechanics of an electric toothbrush involve an intricate balance between design and functionality. The brush heads are usually small, circular, and meant to rotate to effectively remove plaque and bacteria from the teeth. A standard model rotates between predefined angles, like the 70-degree arc mentioned in the exercise.
Electric toothbrushes typically operate by oscillating the brush head back and forth many times per minute. This oscillation translates the angular speed into linear motion, which is vital for effective plaque removal. By oscillating 7600 times per minute, the toothbrush mentioned in the exercise showcases a superior cleaning capability compared to manual brushing.

• The primary component is the motor, which powers the brush head's movement.
• Mechanics include generating enough torque to move the head swiftly while still ensuring it’s not too harsh on dental surfaces.

These technologies make electric toothbrushes particularly effective at maintaining oral health, providing a boost over traditional hand-powered brushing methods.

Radian Conversion

Radian conversion is a fundamental skill when dealing with rotational motion, as many scientific calculations require angles expressed in radians. In our everyday world, we frequently use degrees to measure angles, but radians provide a natural way of quantifying angles in scientific contexts.
In this exercise, we shifted from degrees to radians, as shown in the conversion: 70 degrees was converted to approximately 1.222 radians. To convert degrees to radians, use the factor:

• \( \pi \text{ radians} = 180^{\circ} \)

It helps to remember that one complete rotation (360 degrees) is \( 2\pi \) radians. This relationship is crucial as it allows you to interpret how far an object rotates in terms of its linear path, and is used frequently in physics and engineering to solve a variety of motion-related problems.

Manual vs Electric Brushing

When comparing manual to electric brushing, the differences are largely in efficiency and effectiveness. Manual brushing relies on the user's dexterity to effectively remove plaque. It is estimated that a person can only achieve speeds up to 0.1 \( \text{m/s} \) with manual brushing.
On the other hand, an electric toothbrush significantly enhances these speeds. The exercise showed that an electric brush achieves bristle speeds of around 0.544 \( \text{m/s} \). This increased speed helps the bristles more effectively clear away plaque and food particles.

• Manual brushing requires mindful attention to reach all surfaces thoroughly.
• Electric brushes are advantageous for individuals with limited manual dexterity or impaired fine motor control.

Overall, while both methods require proper technique to maintain oral health, electric toothbrushes offer a more consistent and potentially more effective brushing experience.