How can rigid body have more than 1 speed?

Question

I have a doubt that circular ring is a rigid object and we know every point on the rigid obj must have same velocity or speed. But in case of pure rolling of sphere or others we can find topmost point has velocity 2v lowermost have 0 how's that possible..i am not able to visualise it.

Answer 1

It's not true that "every point on the rigid obj must have same velocity or speed." As you pointed out, if the body is rotating, this is not the case. If the body does have the same velocity at every point, it simply means it is moving in pure translation with no rotation.

Answer 2

You are confusing linear velocity with angular velocity. Consider a circular rigid body rolling in a plane without slipping.

The points in a rotating rigid body do not all have the same linear velocity. Let V be the velocity of the center of mass (CM). As you state, the velocity at the top is 2V, twice the velocity of the center of mass. (And the instantaneous velocity of the point on the surface is 0.) The points in the body have different linear velocities.

It can be shown that in general every point in a rigid body is rotating around every other point in the rigid body with the same unique (instantaneous) angular velocity vector. Advanced mechanics textbooks prove this. This is true for general three-dimensional motion of the rigid body. For the circular body rolling in a plane, the angular velocity of the CM point is V/R where R is the radius of the object and V the velocity of the CM; the angular velocity of the point at the top is 2V/2R = V/R. The angular velocity of the two points is the same.

Answer 3

A rotating rigid body has a different velocity (speed + direction) at each point on the body. Only along the rotation axis, the velocity is the same.

This is often described as a velocity field, with the interpretation of velocity (vector) only being a position function.

In the potato example above the body is rotating about an arbitrary axis through the origin O.

At each point on the potato A, B, C with position vectors $\vec{r}_A$, $\vec{r}_B$ and $\vec{r}_V$ respectively has a different velocity given by the equations

$$ \begin{aligned} \vec{v}_A & = \vec{\Omega} \times \vec{r}_A \\ \vec{v}_B & = \vec{\Omega} \times \vec{r}_B \\ \vec{v}_C & = \vec{\Omega} \times \vec{r}_C \\ \end{aligned}$$

where $\times$ is the vector cross product.

This is a direct result of the property of rigid body rotations which constrain the distances between all points on a body to be fixed. This means for example $d_{\rm AB} = {\rm distance}(\vec{r}_A,\; \vec{r}_B)$ is constant at all time frames throughout the rotation.

Note that if the perpendicular distance to the rotation axis is the same, then the magnitude of the velocity vector (i.e. the speed) of each point is the same. This is to say, every point on the outside of a rotating disk has the same speed. But the velocity is different, since the direction of motion is different for each point.

To generalize this scenario if the origin point O is also translating with a velocity vector $\vec{v}_O$, then this velocity gets added to the velocity of each point due to rotation

The resulting velocity value for each point is:

$$ \begin{aligned} \vec{v}_A & = \vec{v}_O + \vec{\Omega} \times \vec{r}_A \\ \vec{v}_B & = \vec{v}_O +\vec{\Omega} \times \vec{r}_B \\ \vec{v}_C & = \vec{v}_O +\vec{\Omega} \times \vec{r}_C \\ \end{aligned}$$

In this scenario, each point has a different speed as well as different velocity.

Answer 4

. . . . we know every point on the rigid object must have same [linear] velocity . . . is not true; it is the angular velocity which is the same.

Perhaps the simplest rigid body to think about in the context of rotation is a door.
When a door is opened it is clearly the case that all parts of a door do not travel at the same velocity.

The motion of each particle which make up a rotating wheel which is not slipping relative to the ground can be thought of as the sum of the constant rotational motion about the centre of mass of the wheel and the constant horizontal translational motion as shown below.

Note that when these two motions are added together the linear velocity of the particles which make up the wheel are not the same.