Problem 78 - University Physics with Modern Physics

Question

While redesigning a rocket engine, you want to reduce its weight by replacing a solid spherical part with a hollow spherical shell of the same size. The parts rotate about an axis through their center You need to make sure that the new part always has the same rotational kinetic energy as the original part had at any given rate of rotation. If the original part had mass \(M,\) what must be the mass of the new part?

Step-by-Step Solution

Answer

The mass of the new part must be \( \frac{3}{5} M \).

Step 1: Understand the problem

We are replacing a solid spherical part with a hollow spherical shell and we need to ensure the rotational kinetic energies are the same at any rate of rotation. Let the mass of the solid sphere be M.

Step 2: Formula for rotational kinetic energy

The rotational kinetic energy for an object rotating about an axis is given by the formula: \( KE = \frac{1}{2} I \omega^2 \), where \( I \) is the moment of inertia, and \( \omega \) is the angular velocity. Both solid and hollow spheres must have the same \( KE \).

Step 3: Determine the moment of inertia for the solid sphere

The moment of inertia \( I \) for a solid sphere rotating about an axis through its center is \( I_{solid} = \frac{2}{5} MR^2 \).

Step 4: Determine the moment of inertia for the hollow spherical shell

The moment of inertia \( I \) for a hollow spherical shell rotating about an axis through its center is \( I_{hollow} = \frac{2}{3} m R^2 \), where \( m \) is the mass of the hollow shell.

Step 5: Equate the rotational kinetic energies

To ensure the same rotational kinetic energy, equate the two energies: \( \frac{1}{2} I_{solid} \omega^2 = \frac{1}{2} I_{hollow} \omega^2 \). Since \( \omega \) is the same, we can cancel it on both sides, reducing the equation to \( I_{solid} = I_{hollow} \).

Step 6: Solve for the mass of the hollow shell

Set \( \frac{2}{5} MR^2 = \frac{2}{3} m R^2 \) and solve for \( m \). Cancel \( R^2 \) from both sides as it is non-zero: \( \frac{2}{5} M = \frac{2}{3} m \). Divide both sides by 2 and rearrange to find \( m = \frac{3}{5} M \).

Key Concepts

Moment of Inertia

The moment of inertia is a critical concept when understanding rotational motion, akin to mass in linear motion. It essentially measures how difficult it is to change the rotational state of an object. Each different shape and mass distribution have its own formula for the moment of inertia. For rotational systems, the distribution of mass relative to the axis of rotation significantly affects how an object behaves under rotational forces.

In our specific problem, we encounter two types of objects: a solid sphere and a hollow spherical shell. Both of these have distinct moments of inertia due to their mass distributions. Importantly, when equating the rotational kinetic energy of two different bodies, we must consider these mass distributions by using their respective moments of inertia.

• Solid Sphere: The moment of inertia for a solid sphere is \( I = \frac{2}{5} MR^2 \) where \( M \) is the mass and \( R \) is the radius.
• Hollow Spherical Shell: The moment of inertia is \( I = \frac{2}{3} m R^2 \) where \( m \) is the mass and \( R \) is the radius.

This distinction is vital for understanding how they react under the same rotational energy conditions.

Solid Sphere

A solid sphere is a three-dimensional object where all the mass is distributed throughout the volume of the sphere. This even distribution leads to the solid sphere having a particular rotational inertia formula of \( I = \frac{2}{5} MR^2 \).

This formula arises because each particle within the sphere contributes to the inertia depending on its distance from the rotational axis. Because the mass is spread evenly, the resulting calculation reflects the sum of all these contributions. Hence, the moment of inertia is lower when compared to hollow structures, assuming the same mass and radius.

When redesigning components, understanding the moment of inertia in shapes like the solid sphere is crucial. It informs us how a bulk effect of mass distribution can influence the rotational dynamics of machinery parts.

Hollow Spherical Shell

A hollow spherical shell differs from a solid sphere primarily because its mass is distributed only on the surface at equal distance from the center. This construction results in a different moment of inertia for rotational motion.

• For a hollow spherical shell, the moment of inertia is \( I = \frac{2}{3} m R^2 \).

Compared to a solid sphere, the hollow shell’s effective mass is further away from the center, making it take more effort to rotate. This higher moment of inertia is essential for applications where distributing weight differently can affect performance and efficiency.

In practical redesign efforts, like in our exercise, switching to a hollow shell while maintaining similar rotational kinetic energy requires compensating for this higher inertia by reducing mass proportionately, thereby aligning its rotational behavior with that of the original solid sphere.

Angular Velocity

Angular velocity is a fundamental parameter in rotational dynamics, representing how fast an object rotates around an axis. It is typically denoted by the symbol \( \omega \) and is akin to linear velocity, which measures how fast an object moves in a straight line.

In a rotational system, angular velocity remains crucial because it directly influences rotational kinetic energy through the formula \( KE = \frac{1}{2} I \omega^2 \). Both solid and hollow structures in our exercise have their rotational energy explicitly dependent on this value.

The constraint of maintaining equal rotational kinetic energy in different objects requires that any changes in mass or distribution do not affect the overall effect of \( \omega \). Therefore, angular velocity is a key to understanding and solving dynamics problems where inertia and energy conditions are equated. Proper control and understanding of angular velocity ensure that transitioning between different material designs does not impact the operational speed or energy efficiency.