Problem 37 - University Physics with Modern Physics

Question

Find the magnitude of the angular momentum of the second hand on a clock about an axis through the center of the clock face. The clock hand has a length of 15.0 cm and a mass of 6.00 g. Take the second hand to be a slender rod rotating with constant angular velocity about one end.

Step-by-Step Solution

Answer

The angular momentum is approximately \( 4.71 \times 10^{-6} \text{ kg·m}^2/\text{s} \).

Step 1: Understand the Problem

The problem requires us to find the magnitude of the angular momentum for the second hand considered as a slender rod rotating about an axis through the center of the clock. We need to use the properties of the second hand: length of 15 cm and mass of 6 g.

Step 2: Convert Units

The length of the second hand is given in cm, and the mass is given in grams. We need to convert these into meters and kilograms respectively. - Length: 15 cm = 0.15 m - Mass: 6 g = 0.006 kg

Step 3: Calculate Moment of Inertia

For a slender rod of length \( L \) and mass \( m \) rotating about one end, the moment of inertia \( I \) is given by:\[ I = \frac{1}{3} m L^2 \] Substituting the given values:\[ I = \frac{1}{3} \times 0.006 \text{ kg} \times (0.15 \text{ m})^2 = 4.5 \times 10^{-5} \text{ kg·m}^2 \]

Step 4: Calculate Angular Velocity

The second hand completes a full circle in 60 seconds, giving it an angular velocity \( \omega \) of:\[ \omega = \frac{2\pi \text{ radians}}{60 \text{ seconds}} = \frac{\pi}{30} \text{ rad/s} \]

Step 5: Calculate Angular Momentum

The angular momentum \( L \) of the second hand can be calculated using:\[ L = I \cdot \omega \]Substituting the values from previous steps:\[ L = 4.5 \times 10^{-5} \text{ kg·m}^2 \times \frac{\pi}{30} \text{ rad/s} = 4.71 \times 10^{-6} \text{ kg·m}^2/\text{s} \]

Step 6: Solution Verification

We have calculated the angular momentum using the formulas for moment of inertia of a rod and the angular velocity of the second hand correctly. Thus, the computed value should be reliable.

Key Concepts

Moment of Inertia

The moment of inertia is a crucial concept in understanding rotational dynamics. It represents the rotational counterpart to mass in linear motion. Essentially, it's a measure of an object's resistance to changes in its rotation.

• The moment of inertia depends on the distribution of mass in an object and the axis about which it rotates.
• For different shapes and axes, there are different formulas to calculate the moment of inertia. In this exercise, the formula for a slender rod rotating about one end is used.

For a slender rod with mass \( m \) and length \( L \), the moment of inertia \( I \) about one end is calculated as:\[ I = \frac{1}{3} m L^2 \]This formula indicates that both the mass and length of the rod greatly influence its resistance to rotational changes. The longer or heavier the rod, the larger the moment of inertia, meaning it will be harder to spin or stop it from spinning once it starts.

Angular Velocity

Angular velocity describes the speed of rotation of an object around a certain axis. It is the rate at which the angular position or orientation of an object changes with time. The unit of angular velocity is expressed as radians per second (rad/s).

• In the context of this exercise, the second hand of a clock rotates at a consistent speed.
• It completes one full revolution every 60 seconds, reflecting a uniform circular motion.

The angular velocity \( \omega \) can be derived from the formula:\[ \omega = \frac{2\pi}{T} \]where \( T \) is the period of rotation in seconds. For the second hand:\[ \omega = \frac{2\pi}{60} = \frac{\pi}{30} \text{ rad/s} \] This consistent angular velocity simplifies calculations in rotational dynamics, making it a foundational measure in understanding rotational motion behaviors.

Rotational Dynamics

Rotational dynamics is the area of physics examining the effects of torques and angular motion of objects. It extends principles of linear dynamics to rotating objects.

• Key factors include torque, angular momentum, and moments of inertia.
• Rotational dynamics help predict how forces applied to objects will affect their spinning motion.

In this specific exercise, the angular momentum \( L \) is important because it combines the moment of inertia and angular velocity to describe the overall motion:\[ L = I \cdot \omega \]Angular momentum provides insight into the conservation law, where the total angular momentum in a closed system remains constant unless acted upon by an external torque. This principle is a powerful tool for analyzing both simple and complex systems in rotational motion. Understanding these principles enables us to predict and manipulate rotational behaviors in practical applications like mechanical engineering and celestial motions.