Problem 98 - University Physics with Modern Physics

Question

A 55-kg runner runs around the edge of a horizontal turntable mounted on a vertical, frictionless axis through its center. The runner's velocity relative to the earth has magnitude 2.8 \(\mathrm{m} / \mathrm{s}\) . The turntable is rotating in the opposite direction with an angular velocity of magnitude 0.20 rad/s relative to the earth. The radius of the turntable is \(3.0 \mathrm{m},\) and its moment of inertia about the axis of rotation is 80 \(\mathrm{kg} \cdot \mathrm{m}^{2} .\) Find the final angular velocity of the system if the runner comes to rest relative to the turntable. (You can model the runner as a particle.)

Step-by-Step Solution

Answer

The final angular velocity of the system is approximately 0.775 rad/s.

Step 1: Define Angular Momentum Conservation

Since there are no external torques acting on the system, the total angular momentum must be conserved. This means the initial angular momentum of the runner and the turntable will equal the final angular momentum when the runner comes to rest relative to the turntable. Therefore, \( L_{initial} = L_{final} \).

Step 2: Calculate Initial Angular Momentum

Calculate the initial angular momentum of the system. The angular momentum of the turntable is \( L_{turntable} = I_{turntable} \times \omega_{turntable} \), where \( I_{turntable} = 80 \, \mathrm{kg \cdot m^2} \) and \( \omega_{turntable} = 0.20 \, \mathrm{rad/s} \). So, \( L_{turntable} = 80 \, \mathrm{kg \cdot m^2} \times 0.20 \, \mathrm{rad/s} = 16 \, \mathrm{kg \cdot m^2/s} \). Now, the angular momentum of the runner is \( L_{runner} = m_{runner} \times v_{runner} \times r \), where \( m_{runner} = 55 \, \mathrm{kg} \), \( v_{runner} = 2.8 \, \mathrm{m/s} \), and \( r = 3.0 \, \mathrm{m} \). So, \( L_{runner} = 55 \, \mathrm{kg} \times 2.8 \, \mathrm{m/s} \times 3.0 \, \mathrm{m} = 462 \, \mathrm{kg \cdot m^2/s} \). The initial total angular momentum \( L_{initial} = L_{runner} - L_{turntable} = 462 - 16 = 446 \, \mathrm{kg \cdot m^2/s} \).

Step 3: Find Final Angular Velocity

In the final state, the runner comes to rest relative to the turntable, so they have the same angular velocity denoted as \( \omega_{final}\). Therefore, the final angular momentum is given by \( L_{final} = (I_{turntable} + m_{runner} \times r^2) \times \omega_{final} \). With \( r = 3.0 \, \mathrm{m} \), \( I_{runner} = 55 \times 3.0^2 = 495 \, \mathrm{kg \cdot m^2} \). Consequently, \( L_{final} = (80 + 495) \times \omega_{final} = 575 \times \omega_{final} \). Equating initial and final angular momentum, we have \( 446 = 575 \times \omega_{final} \), which gives \( \omega_{final} = \frac{446}{575} \approx 0.775 \mathrm{rad/s} \).

Key Concepts

Moment of Inertia

The moment of inertia (often symbolized as \( I \)) is a measure of how an object's mass is distributed with respect to an axis of rotation. This property determines how much torque is required for a given angular acceleration about that axis. Imagine trying to spin a barbell versus a single dumbbell; the barbell has more of its mass far from the rotational axis, leading to a higher moment of inertia and making it harder to spin. The moment of inertia for a point mass, like the runner in this exercise, about a given axis is calculated as the product of the mass and the square of the perpendicular distance from the axis, expressed as \( I = m imes r^2 \). In our exercise, since the runner is modeled as a particle, this simplifies our calculation and helps in easy understanding.Understanding moment of inertia is key when examining rotational systems where different components may contribute to the system's overall resistance to changes in rotation.

• If the mass is concentrated further from the axis, the moment of inertia increases.
• If the mass is closer to the axis, the moment of inertia decreases.

In our case, knowing the moment of inertia of both the turntable and the runner helps in calculating the total angular momentum, which governs the conservation laws applied here.

Angular Velocity

Angular velocity is a vector quantity that represents how fast an object rotates around an axis. It is like the rotational counterpart to linear velocity. Instead of moving in a straight line, an object under angular motion sweeps an angle over time, and its angular velocity, typically denoted by \( \omega \), tells us how quickly this angle is changing.Angular velocity is measured in radians per second (rad/s), showing how many radians the object rotates in one second. In the given exercise, the runner initially has a perpendicular linear velocity, which contributes to their angular velocity when considering their motion relative to the turntable.

• For the turntable, we already know its initial angular velocity as 0.20 rad/s, rotating opposite to the runner's direction.
• When the runner comes to rest with respect to the turntable, their final shared angular velocity \( \omega_{final} \) needs to be calculated by balancing out the initial angular momentum.

Connecting these dynamic changes to the concept of angular momentum conservation helps determine the new angular velocity of the entire system when states change.

Rotational Kinematics

Rotational kinematics deals with the description of the rotation of a rigid body without considering the forces or torques that cause the motion. It is directly analogous to linear kinematics but focuses on angular quantities. This includes concepts like angular displacement, angular velocity, and angular acceleration, similar to their linear counterparts but involved in circular paths or rotational motion. In the exercise, understanding rotational kinematics can help to conceptualize how the angular velocity of the system changes when the external conditions such as the runner's motion change. Since no external torques are applied, we rely on conservation of angular momentum to predict the final state of the system. The interplay of the runner's kinematics with the motion of the turntable ultimately determines the combined system's final angular velocity. The key takeaway: - When a system undergoes changes in rotation and involves different mass elements, rotational kinematics provides a detailed quantitative description of how these changes manifest. - The principles of rotational kinematics simplify complex interactions by employing concepts already familiar from linear motion, merely applied pivotally around an axis. This ability to translate between linear and angular descriptions is what allows us to solve complex rotational problems like the one presented here.