Twin skaters approach one another as shown in the figure below and lock hands. (a) (b) (a) Calculate their final angular velocity in rad/s, given each had an initial speed of 1.80 m/s relative to the ice. Each has a mass of 70 kg, and their centers of mass are 0.890 m from their locked hands. You may approximate their moments of inertia to be that of point masses at this radius. X What is the angular momentum of an object moving with a linear velocity about a given point? rad/s (b) Compare the initial and final kinetic energy. KE₁ KEf Initially the two skaters have translational motion while after the collision it is purely rotational. X
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 Problem
Twin skaters approach one another as shown in the figure and lock hands.
#### Figure Explanation
- **Diagram (a):** Shows two skaters moving toward each other with linear velocity \(v = 1.80 \, \text{m/s}\).
- **Diagram (b):** Illustrates the skaters after locking hands and rotating with angular velocity \(\omega\).
### Problem Statement
**(a)** Calculate their final angular velocity \(\omega\) in rad/s. Each skater has:
- An initial speed of \(1.80 \, \text{m/s}\).
- A mass of \(70 \, \text{kg}\).
- Centers of mass \(0.890 \, \text{m}\) from their locked hands.
Approximate their moments of inertia as point masses at this radius.
- **Prompt Question:** What is the angular momentum of an object moving with a linear velocity about a given point? (Answer in rad/s)
**(b)** Compare the initial and final kinetic energy.
\[
\frac{KE_i}{KE_f} =
\]
- **Prompt Explanation:** Initially, the two skaters have translational motion, while after the collision, it is purely rotational.
This exercise demonstrates concepts of conservation of angular momentum and kinetic energy transformation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe7370c0b-05cc-48e6-9b0a-29a272be895a%2F144b8a92-7214-43dc-94e9-d302cb6ff4b5%2Fxh7yq5l_processed.png&w=3840&q=75)
Solution:
Given:
m = mass of each skater = 70 kg
v = initial speed of each skater = 1.80 m/s
r = distance between their centre of mass and locked hands = 0.890 m
ω = final angular velocity = ?
KEi = initial kinetic energy of system
KEf = final kinetic energy of system
R = ratio of = ?
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