Problem 43 - Fundamentals of Physics

Question

Two skaters, each of mass \(50 \mathrm{~kg}\), approach each other along parallel paths separated by \(3.0 \mathrm{~m}\). They have opposite velocities of \(1.4 \mathrm{~m} / \mathrm{s}\) each. One skater carries one end of a long pole of negligible mass, and the other skater grabs the other end as she passes. The skaters then rotate around the center of the pole. Assume that the friction between skates and ice is negligible. What are (a) the radius of the circle, (b) the angular speed of the skaters, and (c) the kinetic energy of the two-skater system? Next, the skaters pull along the pole until they are separated by \(1.0 \mathrm{~m}\). What then are (d) their angular speed and (e) the kinetic energy of the system? (f) What provided the energy for the increased kinetic energy?

Step-by-Step Solution

Answer

(a) 1.5 m, (b) 0.933 rad/s, (c) 98 J, (d) 8.397 rad/s, (e) 882 J, (f) Work by skaters.

Step 1: Determine the Radius of the Circle

Initially, the two skaters are separated by a distance of 3.0 m, and they meet at the center of this distance when holding the pole. The radius of the circle about which they rotate is half of this separation.Thus, the radius \( r = \frac{3.0 \space \mathrm{m}}{2} = 1.5 \space \mathrm{m} \).

Step 2: Calculate the Initial Angular Speed

To find the angular speed, we use the formula for linear velocity in terms of angular velocity \( v = \omega r \), where \( v = 1.4 \space \mathrm{m/s} \) and \( r = 1.5 \space \mathrm{m} \).Solving for \( \omega \), we get:\[ \omega = \frac{v}{r} = \frac{1.4 \space \mathrm{m/s}}{1.5 \space \mathrm{m}} = \frac{1.4}{1.5} \approx 0.933 \space \mathrm{rad/s} \]

Step 3: Calculate the Initial Kinetic Energy

The kinetic energy (KE) of the system can be determined using the rotational kinetic energy formula \( KE = \frac{1}{2} I \omega^2 \), where the moment of inertia \( I \) for two masses at radius \( r \) is \( I = 2mr^2 \). Substituting the values:\[ I = 2 \times 50 \times (1.5)^2 = 225 \space \mathrm{kg \, m}^2 \]\[ KE = \frac{1}{2} \times 225 \times (0.933)^2 \approx 98 \space \mathrm{J} \]

Step 4: Calculate Angular Speed After Pulling Closer

After pulling closer, the skaters reduce the distance between them to 1.0 m, giving a new radius of \( r = 0.5 \space \mathrm{m} \). By the conservation of angular momentum \( I_1 \omega_1 = I_2 \omega_2 \), and since \( I \) is proportional to \( r^2 \), we compute:\[ \omega_2 = \omega_1 \times \left( \frac{r_1}{r_2} \right)^2 = 0.933 \times \left( \frac{1.5}{0.5} \right)^2 \]\[ \omega_2 = 0.933 \times 9 = 8.397 \space \mathrm{rad/s} \]

Step 5: Calculate Kinetic Energy After Pulling Closer

Using the new angular speed, calculate the new kinetic energy:\[ I_2 = 2 \times 50 \times (0.5)^2 = 25 \space \mathrm{kg \, m}^2 \]\[ KE = \frac{1}{2} \times 25 \times (8.397)^2 \approx 882 \space \mathrm{J} \]

Step 6: Source of Increased Kinetic Energy

The work done by the skaters in pulling the pole to decrease the distance between them is the source of the increased kinetic energy. This work changes the potential energy into kinetic energy.

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. In the context of rotational dynamics, it refers to the energy associated with the rotation of a body. For the skaters, their motion around the center point creates rotational kinetic energy.
Rotational kinetic energy can be calculated using the formula:

• \( KE = \frac{1}{2} I \omega^2 \)

The Components of Rotational Kinetic Energy

• \( I \) stands for the moment of inertia, which depends on the distribution of the mass around the axis of rotation. For our skaters, each contributes to the moment of inertia depending on how far they are from the pivot.
• \( \omega \) is the angular speed, which is how fast they rotate around the center.

Initial and Final Kinetic Energy

Initially, when they just grab the pole, the skaters have a certain amount of kinetic energy calculated with their initial rotational speed and distance. As they move closer to one another, the distance decreases, leading to changes in the moment of inertia and thus increasing their angular speed and kinetic energy. This shows that when they pull themselves closer, their energy of motion increases significantly, as evidenced by the rise from 98 Joules to approximately 882 Joules.

Rotational Dynamics

Rotational dynamics encompasses the study of rotating bodies and the forces and torques that affect their motion. It plays a critical role in understanding the skaters' behavior as they move around the center of the pole.
Rotational dynamics are similar to linear dynamics, but with a few key differences:
Instead of linear velocity, we use angular velocity \( \omega \).
Instead of linear acceleration, we have angular acceleration \( \alpha \).
Instead of mass \( m \) in linear dynamics, we have the moment of inertia \( I \) in rotational dynamics.

Angular Speed and Motion Around a Circle

When they first meet and begin to rotate, their linear velocities convert into an angular speed, \( \omega \), as they move in a circular path. This is calculated using \( \omega = \frac{v}{r} \), where \( v \) is linear velocity, and \( r \) is the radius of the circle formed by their outstretched arms holding the pole.

Effect of Distance on Rotation

As they pull themselves closer, observe how the change in radius affects their angular velocity. Angular momentum's conservation principle dictates that reducing their separation increases their speed of rotation. This phenomenon is what makes figure skaters spin faster when they pull their arms in.

Conservation Laws

Conservation laws in physics are foundational, indicating that certain properties remain constant through any process. For the skaters, the principle of conservation of angular momentum is particularly relevant.

Conservation of Angular Momentum

Angular momentum \( L \) is given by the product of moment of inertia \( I \) and angular velocity \( \omega \). When no external torques act on the system, angular momentum must remain constant:

• \( L_1 = L_2 \)
• \( I_1 \omega_1 = I_2 \omega_2 \)

For the skaters, this means that when they pull closer together, their moment of inertia decreases, which must be compensated by an increase in their angular speed, \( \omega \), to keep \( L \) unchanged.

Energy Transformation

The skaters also demonstrate energy transformation between forms. The act of pulling each other closer does work on the system. This work, arising from potential energy stored when they reduce the distance, transforms into additional kinetic energy, explaining the observed energy gain without violating conservation laws. This shows how internal forces within a system can transform energy types, satisfying the conservation of both energy and angular momentum.