Rotational Motion and torque problem

[Alche]

1)The torque of the weight of anybody about any vertical axis is 0. Is it always true?

2)A sphere rolls on a horizontal surface. Is there any point of the sphere which has a vertical velocity?

3) When several forces acts on a particle we take the torque of the resultant force. Is this result valid for the force acting on different particles of a body in such a way that their lines of action intersect at a common point ?

4)In pure rolling angular velocity is independent of the distance r from the axis of rotation. Why ?

 

[Eynstone]

1. Not unless the axis passes through the centre of mass.
2.yes
3.yes
4.The angle subtended doesn't depend on r & hence neither does the angular velocity.

 

[Alche]

1. Not unless the axis passes through the centre of mass.
2.yes
3.yes
4.The angle subtended doesn't depend on r & hence neither does the angular velocity.

Can you tell which point has vertical velocity for second question

 

[Ketman]

2)A sphere rolls on a horizontal surface. Is there any point of the sphere which has a vertical velocity?

Suppose the sphere is rolling to the left. Draw a vertical line through the centre. Every point on the sphere to the right of the line is moving upward, every point to the left is moving downward. The only points on the surface that have zero vertical velocity are the point in contact with the ground, and the point directly opposite (the highest point on the surface). Edit: My bad. I should have said, any point along the vertical line will have no vertical velocity, not just the points at the extreme ends of the line.

 

[PJC01]

1) The statement that the torque of the weight of any body about any vertical axis is always 0 is not always true. The torque of an object is the product of the force applied and the distance from the axis of rotation. If the force is applied at a distance of 0 from the axis, the torque will also be 0. However, if the force is applied at a distance from the axis, the torque will not be 0. Therefore, the torque of the weight of a body about a vertical axis will only be 0 if the weight is applied at the axis itself, which is not always the case.

2) Yes, there is a point on the sphere that has a vertical velocity. This is because when a sphere rolls on a horizontal surface, it has both translational and rotational motion. The point of contact between the sphere and the surface will have a zero velocity, but other points on the sphere will have both horizontal and vertical velocities.

3) The torque of the resultant force acting on a particle is valid for different particles of a body as long as their lines of action intersect at a common point. This is because the torque of a force is independent of the point of application, but rather depends on the direction and magnitude of the force and the distance from the axis of rotation. As long as these factors are the same for all the forces acting on the particles, the resultant torque will be valid for the entire body.

4) In pure rolling, the angular velocity is independent of the distance from the axis of rotation because the point of contact between the rolling object and the surface is always at rest. This means that the velocity of the object's center of mass is equal to the velocity of the point of contact, which does not change with distance from the axis. Therefore, the angular velocity, which is the ratio of the linear velocity to the distance from the axis, will also remain constant.