Why do outermost particles on a rotating rigid body gain more kinetic energy?

Question

This is something which I know from my $9^{th}$ standard that on a rigid body rotating about a fixed axis , the velocity of the outermost particle is the maximum and it should be true so that the rigid body stays in one piece but I really don't understand how does that particle gain the highest kinetic energy.

Consider the figure below in which five identical bodies are connected by rods of negligible masses. The structure is hinged at one of the ends and is placed on a smooth horizontal table. And now a force is applied on the second body from the hinge point.

The bodies start rotating about the hinge and thus the external force does a positive work on the rod and also note that the farthest particle gains more kinetic energy . This means that there must be some mechanism responsible for this energy distribution from its point of application (i.e from the second mass) to all the masses.

I think the hinge point is somehow doing this but I am not sure how exactly does it make this happen because it can't do work (negative) on the rod since the point of application of this hinge force is fixed and thus can't take away energy which the rod gains from the work done by the external force.

So can someone explain the Physical mechanism responsible for the transfer of energy from the point of application of external force such that the point farthest from the hinged point gains more of it ?

Answer 1

In a deformable body, you have shearing, extending, and compressing happening to all of the material blobs that comprise the body.

In a rigid body, these types of deformations are assumed not to occur. Theoretically, we say that the body is internally constrained, as the constraints are internal to the material. A simple example of an internal constraint is in an inextensible string. Inextensibility is the internal constraint, and a tension force develops inside the string which is the constraint force. In contrast you have external constraints like your hinge, which forces the rotation to be about a fixed point. An external constraint force develops as the reaction force from the hinge on the body in order to keep that point fixed.

If I make your example into a continuous rigid rod, there are three internal constraints: no extension or compression, no shearing, and no bending. Two internal constraint forces and an internal constraint moment develop: tension/compression force, shear force, and bending moment. These internal loads withstand deformations associated with the internal (rigidity) constraints. Now if you take sub-pieces of your body, you will expose the internal forces and moment. Take a chunk of your rod out towards the tip, say from 9L/10 to L, where L is the length of the rod. You will expose the shear force which is responsible for accelerating the outermost points of the rod. Take another chunk from, say, 0 to L/10, and you should find the shear force to be a little less. This explains the difference in kinetic energies, and the source of power that is responsible for changing the kinetic energy, which is the shear forces acting along the velocities of each material point.

It is a matter of changing your perspective from a global view of the entire body, to one where you take sub-pieces of the body. Newton's laws hold for any rigid body. But if you give me one rigid body, I can always take out a sub-body which is also rigid. Newton's laws equally hold for any cut that you want to take out, but you will expose the internal forces in doing so.

Great question!

Edit: I want to add that the bending moment also supplies power to sub-pieces of the rigid body, as it acts through an angular velocity. The tension/compression force, in contrast, remains powerless.

Answer 2

A possible explanation: For any of the balls, the kinetic energy has the expression $$K = \frac{I \omega^2}{2}$$ Consider the time rate change:

$$ \frac{dK}{dt} = I \alpha \cdot \omega$$

Now since Inertia can be thought of a the 'spread' of the mass distribution, and the time rate change of kinetic energy is dependent on it, we can infer that the particles which are more spread out from the centre of rotation will have contribute most to increasing the energy under action of an external acceleration.

Answer 3

As you would be knowing the every particle during rotation (lets say no linear motion for now) follows circular path. Now zoom in to the particles, every particle is moving on its equipotential surface (no work is done in moving from one point to other in that case). So you see every particle is trying to remain in the equipotential surface and the maximum kinetic energy is not extra because no work is done by the system, all energy is external and the most important, the distribution of energy is provided by coulombic force so that every particle remains in equipotential surface.

You can think like this: Think of the particles that are on the right side of each particle in your image. The particle 1 for instance, has more particles to the right. So it has more mass to right of it that it has to carry with it to complete the circle. So the energy that was distributed is used in overcoming the tension and also gaining its own velocity. On the other hand the last particle has no mass (or tension) on the right so the energy that was distributed is mostly used in gaining velocity.

Now you see the distributed energy is not responsible for the max velocity of the last particle but instead its the mass, the tension or what we call is "Moment of Inertia" who restricted the left particles to move slowly than the last one.

And that is what makes the particle follow the equipotential surface.

Answer 4

The energy transferring mechanism is the internal structure cohesive force of the rigid body. The structural binding force between mass 2 and 3 transfer the force applied on mass 2 to mass 3.

Imagine that the rod is broken between 2 and 3, then all the masses after 3 will not moved.

As mass 2 is trying to move upward due to the external force, the structure of rigid rod beside the mass 2 feels a dragging force upward. Thus, the force transferred to mass 3, and mass 4, ... This cohesive force will interrupt if the structure cannot hold the strength of the tangential dragging force, and breaks.

Answer 5

In reality there will be some bending of the rod like this

There is a tension in the rod that provides the centripetal force that make the particles move in a circle. The tension will vary along the rod, and so will the angles...

The resultant force on a particle, acting perpendicular to the rod, depends on components of the tension (acting both ways in the rod).

On the third particle from the left there are the two components $T_1 \sin\alpha$ and $- T_2\sin\beta$.

However on the fourth particle there is the force $T_2\sin\beta$ and that component of force would be greater than the resultant force on the third.