Centre of mass Momentum problem

[Tanya Sharma]

Homework Statement

Three identical balls are connected by light inextensible strings with each other as shown and rest over a smooth horizontal table .At the moment t=0,ball B is imparted a velocity.Calculate the velocity of A when it collides with ball C.

Homework Equations

The Attempt at a Solution

Since no external force acts on the system,the CM will continue to move at v_0.
Initially CM is present at ball B.It will continue to move upwards with the same velocity.Momentum of the system remains conserved at all times.

After that I am unable to proceed?

 

[ehild]

Homework Statement

Three identical balls are connected by light inextensible strings with each other as shown and rest over a smooth horizontal table .At the moment t=0,ball B is imparted a velocity.Calculate the velocity of A when it collides with ball C.

Homework Equations

The Attempt at a Solution

Since no external force acts on the system,the CM will continue to move at v_0.
Initially CM is present at ball B.It will continue to move upwards with the same velocity.Momentum of the system remains conserved at all times.

After that I am unable to proceed?

The CM will not move with v₀. You find the velocity of the CM from the initial velocities of all balls.
Yes, the momentum is conserved, and what else?

The string is inextensible that means also a condition for the positions and velocities.

ehild

 

[Tanya Sharma]

Sorry... Velocity of CM is \frac{v_0}{3}

Yes I do feel some constraints are there but am unable to find them

 

[ehild]

In what direction will the CM move? And B?
Draw the positions of the balls when A and C just collide. What can you say about the components of the velocities?

ehild

 

[Tanya Sharma]

The CM will continue to move vertically up with velocity \frac{v_0}{3}

 

[Tanya Sharma]

B will move vertically up
A will move vertically up as well as towards C
C will move vertically up as well as towards A

Now what their velocities will be i am unable to comprehend

 

[ehild]

Where are both A and C when they meet? What is their velocity upward (Vy)? The string is inextensible.

ehild

 

[Tanya Sharma]

A and C will be together when they meet .Their velocity will be same as that of B since the strings are inextensible. i.e when A and C meet They have same upward velocity as that of B but at the same time they have equal horizontal velocities as well.

A and C will have equal horizontal velocities all the time so that the net momentum in x direction will be zero.

In that case when they meet all three have same velocity as that of CM in upward direction...Is that so?

 

[ehild]

The magnitude of the horizontal velocities are the same, but the directions are opposite. So all three balls move upward with Vo/3, and balls A and C have also horizontal components of velocity.

What else is conserved during the motion? The strings exert force on the balls, but what can be the net work of the strings for the whole system?

ehild

 

[Tanya Sharma]

What else is conserved during the motion? The strings exert force on the balls, but what can be the net work of the strings for the whole system?
ehild

The strings exerting forces on the balls are internal forces. "Net work of the strings for the whole system" ? I have never applied that concept before...

I feel only momentum is conserved during the motion.

 

[ehild]

What about energy? Is there any external force? Is there any non-conservative force of interaction between the balls?

The normal force from the table does not do any work. The balls experience only tension from the strings. But the tension is in principle an elastic force. If there is some elastic energy during the motion in the strings, at the end when the vertical velocity of all balls are the same, they get relaxed again, as in the initial moment.
So you can take that the energy is conserved and it is kinetic energy.

ehild

 

[Tanya Sharma]

ehild...thank you very much

 

[Kishlay]

The magnitude of the horizontal velocities are the same, but the directions are opposite. So all three balls move upward with Vo/3, and balls A and C have also horizontal components of velocity.

What else is conserved during the motion? The strings exert force on the balls, but what can be the net work of the strings for the whole system?

ehild

no... we want the velocity of ball C relative to ball A when it collides.

 

[ehild]

Kishlay, we are not allowed to give full solutions. Show, what you think about the problem. We spoke about conservation of momentum and conservation of energy. What do they mean in this case?

ehild

 

[Kishlay]

ok ok... i got the answer for this question . thanks...!