Problem 90 - Physics for Scientists and Engineers with Modern Physics

Question

What is the acceleration experienced by the tip of the 1.5-cm-long sweep second hand on your wrist watch?

Step-by-Step Solution

Answer

The acceleration experienced by the tip is approximately \( 1.65 \times 10^{-5} \text{ m/s}^2 \).

Step 1: Identify the Given Variables

The length of the sweep second hand is provided as 1.5 cm, which we will denote as the radius \( r = 1.5 \text{ cm} = 0.015 \text{ meters} \). The second hand completes one full rotation in 60 seconds, which is the period \( T \).

Step 2: Determine the Angular Velocity

The angular velocity \( \omega \) of the second hand can be calculated using the formula \( \omega = \frac{2\pi}{T} \), where \( T \) is the period. Substituting \( T = 60 \text{ seconds} \), we have \( \omega = \frac{2\pi}{60} = \frac{\pi}{30} \text{ rad/s} \).

Step 3: Calculate the Centripetal Acceleration

The formula for centripetal acceleration \( a_c \) is given by \( a_c = \omega^2 \times r \). Substitute \( \omega = \frac{\pi}{30} \) and \( r = 0.015 \text{ m} \) into the equation: \( a_c = \left(\frac{\pi}{30}\right)^2 \times 0.015 \).

Step 4: Simplify to Find the Acceleration

Solving the equation, we get \( a_c = \frac{\pi^2}{900} \times 0.015 = \frac{\pi^2 \times 0.015}{900} \). Calculating this gives \( a_c \approx 1.65 \times 10^{-5} \text{ m/s}^2 \).

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around a point or axis. It is a vector quantity and is often represented by the Greek letter \( \omega \). In the context of rotational motion, angular velocity describes how fast the angular position of an object changes with time. For instance, when you observe the second hand on a traditional wristwatch, it completes one full rotation every 60 seconds. This means that the angular velocity can be calculated using the formula:
\[\omega = \frac{2\pi}{T}\]where:

• \( \omega \) is the angular velocity
• \( T \) is the period, or the time it takes to complete one rotation

In the exercise, the period \( T \) is 60 seconds, leading to an angular velocity of \( \omega = \frac{\pi}{30} \text{ rad/s} \).
Understanding angular velocity is essential for analyzing circular motion, as it helps compare how different objects move around their respective circular paths.

Period of Rotation

The period of rotation is the time taken for one complete revolution around a circular path. It's a crucial concept in understanding rotational dynamics because it allows us to predict and calculate various attributes of an object's motion. The period is denoted by \( T \) and is measured in seconds.
For example, consider the second hand of a watch. It takes exactly 60 seconds to complete one rotation, so the period \( T \) is 60 seconds. This uniform period helps us calculate other aspects of motion, such as angular velocity and, subsequently, centripetal acceleration.
To summarize, a clear grasp of the period of rotation helps us determine how often and how quickly an object can return to its starting position during rotational motion.

Rotational Motion

Rotational motion refers to the motion of an object around a central point or axis. It is how many parts of everyday devices and natural systems operate. When describing rotational motion, several parameters, including angular velocity and the period of rotation, come into play.
In systems undergoing rotational motion, all parts of the object move in circular paths around the axis but at different speeds depending on their distance from the axis. For example, in the wristwatch exercise, every point on the second hand describes a circle, with different aspects being key to understanding its motion:

• Axes of Rotation: These are the points about which the rotational motion occurs.
• Radius of Rotation: This is the distance from the axis to any given point on the rotating object, influencing its linear speed.
• Angular Velocity: This indicates how fast points on the object complete rotational cycles.

A comprehensive understanding of rotational motion hinges on the ability to relate these factors to one another through various formulas and calculations, helping to determine motion's efficiency and effectiveness in practical applications.