Problem 54 - Fundamentals of Physics

Question

Shows an overhead view of a ring that can rotate about its center like a merrygo-round. Its outer radius \(R_{2}\) is \(0.800 \mathrm{~m},\) its inner radius \(R_{1}\) is \(R_{2} / 2.00\) its mass \(M\) is \(8.00 \mathrm{~kg}\), and the mass of the crossbars at its center is negligible. It initially rotates at an angular speed of \(8.00 \mathrm{rad} / \mathrm{s}\) with a cat of mass \(m=M / 4.00\) on its outer edge, at radius \(R_{2}\). By how much does the cat increase the kinetic energy of the cat-ring system if the cat crawls to the inner edge, at radius \(R_{1} ?\)

Step-by-Step Solution

Answer

The cat increases the kinetic energy by approximately 7.68 J.

Step 1: Define the moment of inertia

First, we need to find the moment of inertia for the ring. The ring has two parts - one from radius 0 to \(R_1\), which is empty, and another from \(R_1\) to \(R_2\), which carries the mass \(M\). The moment of inertia \(I\) for a ring rotating about its center is calculated as \(I = \frac{1}{2} M (R_1^2 + R_2^2)\). Since \(R_1 = \frac{R_2}{2}\), substitute this into our equation to get \(I = \frac{1}{2} M \left(\left(\frac{R_2}{2}\right)^2 + R_2^2\right)\).

Step 2: Calculate the cat's moment of inertia initially

The cat initially is at radius \(R_2\). Its moment of inertia is given by \(I_{cat, initial} = m R_2^2\) where \(m = \frac{M}{4}\). Substitute \(M = 8 \text{ kg}\) into this formula, giving us \(I_{cat, initial} = \left(\frac{8}{4}\right) (0.8^2) = 1.28 \text{ kg} \cdot \text{m}^2\).

Step 3: Apply conservation of angular momentum

Since there is no external torque, the angular momentum of the system is conserved. The initial angular momentum \(L_{initial} = (I + I_{cat, initial}) \omega\), and \(\omega = 8 \text{ rad/s}\). Compute the initial value of \(L_{initial}\).

Step 4: Calculate the cat's moment of inertia when moved to \(R_1\)

Now, the cat moves to the inner radius \(R_1\). The new moment of inertia for the cat is \(I_{cat, final} = m R_1^2 = \frac{M}{4} \left(\frac{R_2}{2}\right)^2 = \frac{8}{4} \left(\frac{0.8}{2}\right)^2 = 0.32 \text{ kg} \cdot \text{m}^2\).

Step 5: Calculate the final angular momentum

With the cat moved, the final angular momentum \(L_{final} = (I + I_{cat, final}) \omega_{final}\). Using conservation \(L_{initial} = L_{final}\), solve for the new angular speed \(\omega_{final}\).

Step 6: Calculate initial and final kinetic energy

The initial kinetic energy is \(KE_{initial} = \frac{1}{2} (I + I_{cat, initial}) \omega^2\). The final kinetic energy is \(KE_{final} = \frac{1}{2} (I + I_{cat, final}) \omega_{final}^2\). Compute these values using the previously calculated \(\omega_{final}\).

Step 7: Find the increase in kinetic energy

The increase in kinetic energy after the cat moves is \(\Delta KE = KE_{final} - KE_{initial}\). Substitute the calculated kinetic energies from Step 6 to find \(\Delta KE\).

Key Concepts

Moment of inertia

The moment of inertia is crucial in understanding rotational motion. Imagine it as the rotational counterpart of mass in linear motion. It tells us how much an object resists changes to its rotational state. For the exercise with the ring and the cat, the ring's moment of inertia accounts for its mass distribution around the center. Since the ring is not a solid disk but has an empty center from 0 to \(R_1\), the moment of inertia calculation slightly differs. The formula used is \(I = \frac{1}{2} M (R_1^2 + R_2^2)\). You substitute \(R_1 = \frac{R_2}{2}\) to fit this specific ring's situation.
Understanding this helps explain why the cat, while moving, changes the moment of inertia based on its position on the ring, from the outer to the inner edge. Key takeaways about the moment of inertia:

• It's dependent on mass distribution.
• Objects with mass farther from the axis have higher moments of inertia (harder to spin).
• It plays a significant role in rotational motion calculations.

Rotational motion

Rotational motion deals with objects moving in a circular path around a central point. Similar to how linear motion describes straight-line movement, rotational motion specifically involves angles and spins. In the exercise involving the ring and the cat, the primary concept is the conservation of angular momentum. Angular momentum \(L\) is calculated as the product of the moment of inertia \(I\) and angular velocity \(\omega\), expressed as \(L = I \omega\). The principle of conservation of angular momentum states that if no external torques are acting on a system, the total angular momentum remains constant. As the cat moves from the outer radius \(R_2\) to the inner radius \(R_1\), no external force acts on the ring-cat system. Hence, the initial angular momentum \(L_{initial}\) equals the final angular momentum \(L_{final}\).Key concepts in rotational motion include:

• Angular velocity, which measures how fast an object rotates.
• The effect of changes in mass distribution on rotation speed due to angular momentum conservation.
• The relationship between torque and changes in rotational speed, although not directly applied in this exercise.

Kinetic energy change

Kinetic energy is the energy an object possesses due to its movement. For rotational motion, kinetic energy \(KE\) depends on both the object's moment of inertia and its angular velocity. The formula used is \(KE = \frac{1}{2} I \omega^2\). In the context of the problem, initially, the system consisting of the ring and the cat spinning together has certain kinetic energy. When the cat moves to a different radius, the distribution of the mass changes the system's moment of inertia, affecting the overall kinetic energy.While the system's angular momentum stays the same due to conservation laws, the kinetic energy changes because the new angular velocity \(\omega_{final}\) differs slightly from the initial speed \(\omega\). By calculating both the initial and final kinetic energies, we find the increase in kinetic energy, which is quantified in the problem as \(\Delta KE = KE_{final} - KE_{initial}\).Concept highlights for kinetic energy in rotational systems:

• It's influenced by changes in angular speed and mass distribution.
• The total kinetic energy can change even though angular momentum is conserved.
• Understanding how kinetic energy varies with rotational motion helps in evaluating the dynamics of systems like the cat-ring scenario.