Centre of mass Momentum problem

In summary: The conservation of momentum means that the total momentum of the system is conserved at all times. The conservation of energy means that the total energy of the system is conserved at all times. So in this situation, the momentum and energy are both conserved.
  • #1
Tanya Sharma
1,540
135

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 [itex]v_0[/itex].
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?
 

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  • #2
Tanya Sharma said:

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 [itex]v_0[/itex].
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
 
  • #3
Sorry... Velocity of CM is [itex]\frac{v_0}{3}[/itex]

Yes I do feel some constraints are there but am unable to find them
 
  • #4
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
 
  • #5
The CM will continue to move vertically up with velocity [itex]\frac{v_0}{3}[/itex]
 
  • #6
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
 
  • #7
Where are both A and C when they meet? What is their velocity upward (Vy)? The string is inextensible.

ehild
 

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  • #8
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?
 
  • #9
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
 
  • #10
ehild said:
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.
 
  • #11
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
 
  • #12
ehild...thank you very much
 
  • #13
ehild said:
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.
 
  • #14
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
 
  • #15
ok ok... i got the answer for this question . thanks...!
 

1. What is the center of mass momentum problem?

The center of mass momentum problem is a physics concept that deals with the motion and momentum of objects that have a center of mass. It involves calculating the total momentum of a system of objects and determining how it changes over time.

2. How do you calculate the center of mass of an object?

To calculate the center of mass of an object, you need to find the average position of all the particles that make up the object. This can be done by finding the weighted average of the position of each particle, where the weight is the mass of the particle.

3. What is the difference between center of mass and center of gravity?

The center of mass and center of gravity are often used interchangeably, but they have slightly different meanings. The center of mass is the average position of all the particles in an object, while the center of gravity is the point where the force of gravity acts on an object. In most cases, the two are very close to each other, but they can differ in situations where there is a significant gravitational field gradient.

4. How does the center of mass affect the motion of an object?

The center of mass plays a crucial role in determining the motion of an object. The laws of motion state that the center of mass of an isolated system will remain at rest or in a constant velocity unless acted upon by an external force. This means that the center of mass affects the overall motion of an object and can help predict how the object will move.

5. What are some real-life applications of the center of mass momentum problem?

The center of mass momentum problem has many practical applications in fields such as engineering, physics, and astronomy. It is used to design stable structures, analyze the motion of satellites and spacecraft, and understand the behavior of celestial bodies. It is also essential in understanding the movement of objects in sports, such as balancing a bicycle or calculating the trajectory of a ball in motion.

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