Two balls and a rod colliding with a ball

[Jahnavi]

Homework Statement

Homework Equations

The Attempt at a Solution

The length of the rod is along
y direction and (A+B+rod) is traveling along +x direction .

Since there is no external force along x direction momentum is conserved along x direction . If v' is velocity of CM of (A+B+P) then

2mv = 3mv' gives v' = 2v/3 .

Suppose O is a point on the table just below the new CM at the moment of collision . Now , I would like to conserve angular momentum about O .

My doubt is regarding the force exerted by the rod on the balls . If the force exerted by rod is along it's length then angular momentum can be conserved .

But is it correct to assume that the rod exerts a force only along its length ? Has it anything to do with the rod being light ?

 

[haruspex]

My doubt is regarding the force exerted by the rod on the balls .

That force is internal to the system.
In fact it will be along the rod's length for the reason you give. Otherwise the forces on the rod would not cancel completely; there would be either a net force or a net torque. Since the rod is treated as massless that would resukt in an infinite acceleration.

 

[Jahnavi]

Thanks .

What if the rod has some mass ? Would the force exerted by the rod on the balls be along the length ?

 

[haruspex]

Thanks for replying .

What if the rod has some mass ? Would the force exerted by the rod on the balls be along the length ?

If the rod has mass then there will be transverse forces between it and the balls. But since these are internal to the system angular momentum is still conserved. You just have to include the moment of inertia of the rod in the equation.
By the way, it is often simpler to avoid finding the mass centre of the system. E.g. you could take your axis as the initial location of P.

 

[Jahnavi]

If the rod has mass then there will be transverse forces between it and the balls.

You mean force perpendicular to the length of rod ?

But why would the force by the rod on the two balls be equal .If they are equal only then the net torque will be zero .Only then angular momentum will be conserved ?

If rod has mass then if we consider ball B as our system , then momentum of ball is no longer conserved as there would be a force exerted by rod on the ball along its velocity ?

But if the rod is massless then force has to be along the length .Right ?

 

[haruspex]

You mean force perpendicular to the length of rod ?

Yes.

why would the force by the rod on the two balls be equal

I did not say it would be. If the rod has mass then there will be a net torque to get it rotating and a net linear force to change its velocity.

if we consider ball B as our system

Why would you do that? To answer the posted question, consider the rod and all three balls as the system.

if the rod is massless then force has to be along the length .Right ?

Yes.

 

[Jahnavi]

If the rod has mass then there will be a net torque to get it rotating and a net linear force

And that net torque and linear force on the rod will be provided by the balls in a direction perpendicular to its length .This justifies the existence of force by the rod on the balls in transverse direction unlike the problem in OP . Right ?

Why would you do that? To answer the posted question, consider the rod and all three balls as the system.

If we know that the direction of force applied by the rod on the balls is along its length , then this problem becomes very easy .We don't even require to conserve angular momentum :)

This is one of the reasons why I am thinking about direction of force exerted by the rod on the ball .

So , in those types of problems where there is rod having mass with two equal masses at its end , rotating and translating , the force exerted by rod is not necessarily along its length .

Suppose there was a taut string instead of the rod , the approach and the answers to this problem would remain same . Is that correct ?

 

[ehild]

Now , I would like to conserve angular momentum about O .

My doubt is regarding the force exerted by the rod on the balls . If the force exerted by rod is along it's length then angular momentum can be conserved .

But is it correct to assume that the rod exerts a force only along its length ? Has it anything to do with the rod being light ?

You are right, the angular momentum of the system is conserved. From the text of the problem, the geometry should like as it shown in the picture.

A and P collide, and they exert force on each other during the collision. At this moment, the rod exerts force on the balls A and B in perpendicular directions, but you need not bother about it. Consider A, B, and the rod as a rigid body and find the angular momentum of the system before and after collision,

 

[Jahnavi]

Beautiful picture !

Thanks .

At this moment, the rod exerts force on the balls A and B in perpendicular directions, but you need not bother about it.

But then there would be a net torque on a massless rod producing infinite angular acceleration ?

Consider A, B, and the rod as a rigid body and find the angular momentum of the system before and after collision,

If we know that the direction of force applied by the rod on the balls is along its length, then this problem becomes very easy .We don't even require to conserve angular momentum :)

 

[ehild]

But then there would be a net torque on a massless rod producing infinite angular acceleration ?
If we know that the direction of force applied by the rod on the balls is along its length, then this problem becomes very easy .We don't even require to conserve angular momentum :)

The balls are fixed to the rod. The rod can not have different angular acceleration as the balls. They make a rigid body.
In case of a rigid body we need not consider the forces acting between its parts.
Also, we do not know what forces act during a collision. Therefore, we work with conservation of momentum and conservation of angular momentum of the system consisting of the balls A. B, fixed to the rod, and the ball P.

 

[Jahnavi]

Thank you ehild .

@haruspex , please reply to post#7 .

 

[Jahnavi]

This is a problem from a recent thread at PF .I do not intend to solve this problem . This is just to highlight the issue whether the massless rod can exert transverse force or not .

In the last line the question states that the massless rod exerts sidewise forces.

How is this possible ?

 

[haruspex]

This justifies the existence of force by the rod on the balls in transverse direction unlike the problem in OP . Right ?

Yes.
Note that this says that with a massless rod the linear velocity of ball B should not immediately change. Using a whole system approach that is what I found.

the last line the question states that the massless rod exerts sidewise forces.

Yes, I also saw that remark and could not see how it could be true. What is true is that once the system is in motion the rotation will alter the force along the rod.
Consider the case where the mass on the y-axis is below the x axis. We can now replace the rod with a string. This cannot exert any transverse force, yet the string will be taut and the system behave in the same way.