Rolling/non-rolling motion of unsymmetrical bodies

[raghavgupta93]

I have a sphere (mass m, radius r) whose centre of mass is r/2 distance away from its geometric centre. If it is kept on a smooth horizontal surface with the centre of mass at the same horizontal level as the geometric centre, what exactly is going to happen?

Also, I need to find out the velocity of the COM as the sphere turns by an angle (say, theta).

Given the the moment of inertia of the sphere about an axis passing through its COM and perpendicular to the plane of motion is mr²/3.

It's very tricky to me when it's NOT given that the sphere rolls without sliding (most problems related to unsymmetrical bodies have this mentioned).

 

[LightHero]

In such a case, the sphere will not move at all. It will remain stationary and its centre of mass will remain at the same horizontal level as the geometric centre. The only motion the sphere will experience will be a rotational motion about the centre of mass due to the unbalanced moment of inertia. The angular velocity of the sphere will be determined by the equation of conservation of angular momentum. The angular velocity (ω) will be equal to the torque (τ) divided by the moment of inertia (I). For this case, the torque is equal to the product of the mass (m) and the distance between the COM and the geometric centre (r/2), and the moment of inertia is mr2/3. Therefore, the angular velocity of the sphere will be given by: ω = (mr/2) / (mr2/3) = 3/4 radians/second.

 

[astrokreng]

Based on the information provided, the sphere will experience a combination of rolling and sliding motion as it turns by an angle theta. The center of mass of the sphere will move in a circular path, while the geometric center will slide along the surface. The exact motion of the sphere will depend on the initial conditions and external forces acting on it.

To calculate the velocity of the center of mass, we can use the equation v = ωr, where v is the velocity of the center of mass, ω is the angular velocity, and r is the distance between the center of mass and the axis of rotation. In this case, r/2 would be the distance between the center of mass and the axis of rotation.

To find the angular velocity, we can use the equation τ = Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration. In this case, the torque would be caused by the force of gravity and any other external forces acting on the sphere. The moment of inertia of the sphere can be calculated using the given equation mr2/3.

Overall, the exact motion and velocity of the sphere would require further analysis and calculation based on the specific conditions and forces acting on the system. It is important to note that the presence of unsymmetrical bodies can make the motion more complex and require a more detailed analysis.