To determine the maximum angular velocity of the compound sphere when it is slightly disturbed, we need to analyze the system's dynamics, particularly focusing on the moment of inertia and the forces acting on it. The compound sphere consists of a hemispherical shell and a solid hemisphere, both with the same radius \( r \) and mass \( m \). Let's break down the problem step by step.
Understanding the Components
The compound sphere is formed by two parts:
- Hemispherical Shell: This is a hollow hemisphere with mass \( m \) and radius \( r \).
- Solid Hemisphere: This is a solid hemisphere with the same mass \( m \) and radius \( r \).
Calculating the Moment of Inertia
To find the maximum angular velocity, we first need to calculate the moment of inertia for both components about the axis of rotation. The total moment of inertia \( I \) of the compound sphere will be the sum of the moments of inertia of the shell and the solid hemisphere.
Moment of Inertia of the Hemispherical Shell
The moment of inertia \( I_{\text{shell}} \) of a hemispherical shell about its flat face is given by:
I_{\text{shell}} = \frac{2}{3} m r^2
Moment of Inertia of the Solid Hemisphere
The moment of inertia \( I_{\text{solid}} \) of a solid hemisphere about its flat face is given by:
I_{\text{solid}} = \frac{2}{5} m r^2
Combining the Moments of Inertia
Now, we can find the total moment of inertia \( I_{\text{total}} \) of the compound sphere:
I_{\text{total}} = I_{\text{shell}} + I_{\text{solid}} = \frac{2}{3} m r^2 + \frac{2}{5} m r^2
Finding a Common Denominator
To combine these fractions, we need a common denominator, which is 15:
I_{\text{total}} = \frac{10}{15} m r^2 + \frac{6}{15} m r^2 = \frac{16}{15} m r^2
Angular Velocity and Dynamics
When the compound sphere is slightly disturbed, it will start to rotate. The maximum angular velocity \( \omega_{\text{max}} \) can be derived from the conservation of energy or dynamics of rotational motion. Assuming no external torques act on the system, the angular momentum before and after the disturbance remains constant.
Using Energy Conservation
When the sphere is disturbed, it will convert potential energy into rotational kinetic energy. The potential energy associated with the slight disturbance can be equated to the rotational kinetic energy:
PE = KE
For small displacements, we can assume the potential energy is proportional to the height of the center of mass and the gravitational force acting on it. The rotational kinetic energy is given by:
KE = \frac{1}{2} I_{\text{total}} \omega^2
Final Calculation
Setting the potential energy equal to the kinetic energy allows us to solve for the maximum angular velocity:
mgh = \frac{1}{2} I_{\text{total}} \omega_{\text{max}}^2
Solving for \( \omega_{\text{max}} \) gives:
\omega_{\text{max}} = \sqrt{\frac{2mgh}{I_{\text{total}}}}
In this case, \( h \) would depend on the specific geometry and displacement of the sphere, but the key takeaway is that the maximum angular velocity is directly related to the moment of inertia and the height of the center of mass. The smoother the surfaces, the less friction will affect the motion, allowing for a higher angular velocity.
In summary, by calculating the moment of inertia for both components and applying the principles of rotational dynamics, we can derive the maximum angular velocity of the compound sphere when disturbed. This approach not only highlights the importance of understanding the components of the system but also emphasizes the interplay between energy conservation and rotational motion.
