Question

A uniform spherical asteroid of radius $R_{\mathrm{o}}$ is spinning with angular velocity ${\omega }_{\mathrm{o}}$. As the aeons go by, it picks up more matter until its radius is $R$. Assuming that its density remains the same and that the additional matter was originally at rest relative to the asteroid (anyway on average), find the asteroid's new angular velocity. (You know from elementary physics that the moment of inertia is $\frac{2}{5}M{R}^2$.) What is the final angular velocity if the radius doubles?

Short answer

When the radius doubles, the angular velocity becomes $\frac{{\omega }_{\mathrm{o}}}{4}$.

Step-by-step solution

Identify Initial Conditions

We begin with an asteroid of initial radius $R_{\mathrm{o}}$ and angular velocity ${\omega }_{\mathrm{o}}$. The density is constant, which implies that its mass $M_{\mathrm{o}}$ is proportional to the volume, i.e., $M_{\mathrm{o}}=\rho \cdot \frac{4}{3}\pi R_{\mathrm{o}}^3$ for some constant density $\rho$.

Understand Moment of Inertia

The moment of inertia $I$ of a sphere with mass $M$ and radius $R$ is given by $I=\frac{2}{5}M{R}^2$. We will use this formula to find the initial and final moments of inertia, $I_{\mathrm{o}}$ and $I$, respectively.

Apply Conservation of Angular Momentum

Angular momentum $L$ is conserved. Initially, $L_{\mathrm{o}}=I_{\mathrm{o}}\cdot {\omega }_{\mathrm{o}}=\frac{2}{5}{M}_{\mathrm{o}}{R}_{\mathrm{o}}^2\cdot {\omega }_{\mathrm{o}}$. After acquiring more mass, the final mass $M$ and therefore the new angular momentum is $L=I\cdot \omega =\frac{2}{5}M{R}^2\cdot \omega$. Thus, $\frac{2}{5}{M}_{\mathrm{o}}{R}_{\mathrm{o}}^2\cdot {\omega }_{\mathrm{o}}=\frac{2}{5}M{R}^2\cdot \omega$.

Solve for Final Angular Velocity

Since $M$ remains proportional to the asteroid's volume with constant density, $M=\rho \cdot \frac{4}{3}\pi R^3$. Therefore, $M_{\mathrm{o}}=\rho \cdot \frac{4}{3}\pi R_{\mathrm{o}}^3=\frac{M{R}_{\mathrm{o}}^3}{R^3}$. Substitute $M_{\mathrm{o}}$ in the conservation equation to obtain: $\omega ={\omega }_{\mathrm{o}}\cdot {\left(\frac{R_{\mathrm{o}}}{R}\right)}^2$.

Evaluate for Given Problem Condition

Given that the radius doubles, $R=2R_{\mathrm{o}}$. Substitute into $\omega ={\omega }_{\mathrm{o}}\cdot {\left(\frac{R_{\mathrm{o}}}{R}\right)}^2$ to find $\omega ={\omega }_{\mathrm{o}}\cdot {\left(\frac{R_{\mathrm{o}}}{2R_{\mathrm{o}}}\right)}^2={\omega }_{\mathrm{o}}\cdot \frac{1}{4}$. Thus, when the radius doubles, the final angular velocity is $\omega =\frac{{\omega }_{\mathrm{o}}}{4}$.

Key concepts

Conservation of Angular Momentum

The principle of conservation of angular momentum is a fundamental concept in physics. It states that if no external torque acts on a system, the total angular momentum of the system remains constant. Angular momentum is the rotational equivalent of linear momentum and is given by the product of an object's moment of inertia and its angular velocity. In mathematical terms, this is expressed as $L=I\cdot \omega$. For the asteroid scenario, since there are no external torques acting on it, the initial angular momentum, $L_{\mathrm{o}}$, must equal the final angular momentum, $L$. This allows us to set up an equation that we can solve for the asteroid's new angular velocity after it has accumulated more mass. This relationship is crucial to solving problems that involve rotational dynamics without external torques.

Moment of Inertia

Moment of inertia is a key concept when dealing with rotational dynamics. It measures an object's resistance to changes in its rotation and depends on both its mass and how that mass is distributed with respect to the axis of rotation. For a solid sphere, the moment of inertia $I$ is given by $\frac{2}{5}M{R}^2$, where $M$ is the mass and $R$ is the radius. In the asteroid problem, knowing how to calculate the moment of inertia is vital for determining both the initial and the new angular momentum of the asteroid. By understanding how moment of inertia changes with mass and radius, you can apply this knowledge to find the asteroid's changed rotational state after it accretes more mass.

Angular Velocity

Angular velocity represents how fast an object rotates or spins and is typically measured in radians per second. It is a vector quantity, which means it has both magnitude and direction. In the question involving the spherical asteroid, the initial angular velocity, ${\omega }_{\mathrm{o}}$, is provided, and the task is to find the final angular velocity after the asteroid acquires more mass and its radius increases. With the principle of conservation of angular momentum, the relationship between initial and final angular velocity can be expressed mathematically, showing how changes in the asteroid's properties affect its rotational speed. This concept is particularly important when considering rotational motion in a vacuum or space, where external influences are minimal.

Spherical Asteroid

A spherical asteroid provides an ideal example of symmetry in physics, simplifying the analysis of its properties. Assuming the asteroid remains spherical as it gains mass means its density stays uniform, simplifying the calculation of its mass increase. This fact is important because the moment of inertia formula $\frac{2}{5}M{R}^2$ and any related calculations depend on the object's shape. For any changes in the asteroid, consistent logic is applied: by keeping the shape uniform, we can consistently predict how mass—and thereby volume—increase affects rotational dynamics. This hypothetical scenario helps students understand the systematic approach to solving real-world physics problems involving celestial bodies.

Physics Problem Solving

When tackling physics problems, like the one involving the asteroid, a structured and strategic approach is essential. Here’s a simple guide:

• Identify what you know and what you need to find out—list known quantities such as initial angular velocity, radius, and constraints like constant density.
• Understand the core concepts that apply—use your understanding of conservation of angular momentum, moment of inertia, and angular velocity.
• Set up relevant equations based on these principles—like equating the initial and final angular momentum to find the new angular velocity.
• Solve systematically, step by step, and check units at each stage to avoid mistakes.
• Review the solution to ensure it makes sense physically—consider if, for instance, an increase in mass should logically slow down the rotation as predicted.

Solid problem-solving strategies not only help you solve individual exercises but also develop skills to tackle more complex physics challenges in the future.