Twin skaters approach one another as shown in the figure below and lock hands. (a) Calculate their final angular velocity, given each had an initial speed of 2.10 m/s relative to the ice. Each has a mass of 84.0 kg, and each has a center of mass located 0.760 m from their locked hands. You may approximate their moments of inertia to be that of point masses at this radius. rad/s (b) Compare the initial kinetic energy and final kinetic energy. O tritial kinetic energy is the same as final kinetic energy. O tritial kinetic energy is less than final kinetic energy. O Iritial kinetic energy is greater than final kinetic energy.
Angular Momentum
The momentum of an object is given by multiplying its mass and velocity. Momentum is a property of any object that moves with mass. The only difference between angular momentum and linear momentum is that angular momentum deals with moving or spinning objects. A moving particle's linear momentum can be thought of as a measure of its linear motion. The force is proportional to the rate of change of linear momentum. Angular momentum is always directly proportional to mass. In rotational motion, the concept of angular momentum is often used. Since it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics. To understand the concept of angular momentum first we need to understand a rigid body and its movement, a position vector that is used to specify the position of particles in space. A rigid body possesses motion it may be linear or rotational. Rotational motion plays important role in angular momentum.
Moment of a Force
The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.
![### Twin Skaters Angular Velocity and Kinetic Energy Analysis
**Scenario**: Twin skaters approach one another as shown in the figure below and lock hands.
![Diagram showing twin skaters approaching each other and eventually locking hands]
_A detailed diagram illustrating two skaters approaching each other. The direction of their motion and the position where they lock hands are indicated._
**Problem Statement**:
1. **Calculate Final Angular Velocity**:
- Each skater has an initial speed of \(2.10 \, \text{m/s}\) relative to the ice.
- Each skater has a mass of \(84.0 \, \text{kg}\).
- The center of mass for each skater is located \(0.750 \, \text{m}\) from their locked hands.
- Approximate their moments of inertia as that of point masses at this radius.
2. **Compare Initial and Final Kinetic Energy**:
a. Initial kinetic energy is the same as final kinetic energy.
b. Initial kinetic energy is less than final kinetic energy.
c. Initial kinetic energy is greater than final kinetic energy.
**Answers**:
(a) Calculation of final angular velocity \( \omega_f \):
\[
\omega_f = \frac{}{\text{rad/s}}
\]
(b) Comparison of kinetic energy:
- \( \textcircled{a} \) Initial kinetic energy is the same as final kinetic energy.
- \( \textcircled{b} \) Initial kinetic energy is less than final kinetic energy.
- \( \textcircled{c} \) Initial kinetic energy is greater than final kinetic energy.
**Graph/Diagram Explanation**:
- _The depicted diagram shows the motion paths and positions of the skaters as they move towards each other._
- _It helps visualize the change in position and orientation of the skaters from the point they begin their motion to the point where they lock hands._
This exercise aims to enhance understanding of angular velocity and kinetic energy in a rotational system among students.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb8ba6cb-f074-43c1-b7bc-05cc004f9981%2F8c7f3555-b490-4064-b221-8a586d1fc2b7%2F6m3904p_processed.png&w=3840&q=75)
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