Problem 70 - College Physics

Question

An ice skater has a moment of inertia of \(100 \mathrm{~kg} \cdot \mathrm{m}^{2}\) when his arms are outstretched and a moment of inertia of \(75 \mathrm{~kg} \cdot \mathrm{m}^{2}\) when his arms are tucked in close to his chest. If he starts to spin at an angular speed of 2.0 rps (revolutions per second) with his arms outstretched, what will his angular speed be when they are tucked in?

Step-by-Step Solution

Answer

The skater's angular speed when his arms are tucked in is 2.67 rps.

Step 1: Identify the Problem Type

This problem involves conservation of angular momentum. Since no external torques are acting on the system, the angular momentum will remain constant as the skater changes his arm position.

Step 2: Write the Conservation of Angular Momentum Equation

The formula for angular momentum is \(L = I \cdot \omega\), where \(I\) is the moment of inertia and \(\omega\) is the angular speed. Conservation of angular momentum states that initial angular momentum \(L_i\) is equal to final angular momentum \(L_f\). Thus, \(I_i \cdot \omega_i = I_f \cdot \omega_f\).

Step 3: Substitute Known Values

Given that \(I_i = 100 \; \mathrm{kg} \cdot \mathrm{m}^2\), \(\omega_i = 2.0 \; \mathrm{rps}\), and \(I_f = 75 \; \mathrm{kg} \cdot \mathrm{m}^2\), substitute these values into the conservation equation: \(100 \cdot 2.0 = 75 \cdot \omega_f\).

Step 4: Solve for Final Angular Speed \(\omega_f\)

Rearrange the equation from Step 3 to find \(\omega_f\): \(\omega_f = \frac{100 \cdot 2.0}{75}\). Calculate this to find \(\omega_f = \frac{200}{75} = 2.67 \; \mathrm{rps}\).

Key Concepts

Moment of Inertia

In physics, the moment of inertia is an important concept when dealing with rotational motion. It's a measure of an object's resistance to any change in its state of rotation. Think of it like mass in linear motion, but for spinning objects.

A higher moment of inertia means more effort is needed to change the rotational speed of an object. For the ice skater, when arms are spread out, the mass distribution is further from the rotation axis, making the moment of inertia larger. Conversely, when the skater pulls the arms in, the mass distribution is closer to the axis, reducing the moment of inertia.

Key things to remember about moment of inertia:

• Depends on the mass and how that mass is distributed relative to the axis.
• Units of moment of inertia are \( \mathrm{kg} \cdot \mathrm{m}^{2} \).
• Affects how easily an object can start or stop rotating.

Physics Problem Solving

Solving physics problems can feel like piecing together a puzzle. Starting with identifying what type of problem you're facing helps to choose the correct formulas and principles to use, such as recognizing if conservation laws apply.

Here’s a simple approach to tackle problems involving rotational motion:

• Clearly identify what is given and what needs to be found.
• Recognize which physical principles are relevant, like conservation laws.
• Write the relevant equations that connect the knowns to the unknowns.
• Substitute known values into the equation and solve for the unknown.
• Check if the units are consistent and if the answer is reasonable.

Using these steps not only helps solve the problem but also builds understanding. With practice, these steps become almost like a natural instinct when tackling physics problems.

Conservation Laws in Physics

Conservation laws play a crucial role in physics, acting as guiding principles that can simplify problem-solving in many complex systems. One of these fundamental laws is the conservation of angular momentum.

Angular momentum is conserved when there is no external torque acting on a system. This means the initial and final angular momentum of a system are equal, provided the system is closed. In the example of the ice skater:

• Initial angular momentum is calculated using moment of inertia and initial angular speed.
• As the skater pulls arms in, the decrease in moment of inertia results in an increase in angular speed to maintain constant angular momentum.
• Formula used: \( I_i \cdot \omega_i = I_f \cdot \omega_f \)

Understanding this conservation allows us to predict changes in rotation when the distribution of mass changes, like the skater spinning faster as arms are drawn inward.

These laws are useful not just theoretically, but practically, influencing fields from astrophysics to engineering and beyond.