Force in Center of Mass frame equaling zero

In summary: My apologies, I made a mistake in my previous reply. The correct formula would be:F - Ffict = dP/dtwhere Ffict is the fictitious force acting on the particles in the accelerated frame, and it is equal to -M1a and -M2a.So the equation would be:F - M1a - M2a = dP/dtAnd because a = g in this case, the equation becomes:F - M1g - M2g = dP/dtwhich is equal to 0, since the change in momentum is equal to 0 in the CM frame.In summary, the total force acting on the masses in the center of mass frame is
  • #1
CGandC
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Homework Statement



Suppose I have particles with masses m1 and m2 , both are in freefall:
upload_2017-9-26_21-30-7.png

Then , looking at the center of mass coordinates , I know that the sum of momentums of masses m1 and m2 in the center of mass coordinates is equal to zero : P1c + P2c = 0

so now I know that the total force acting on the masses in the center of mass frame is :
Fcmf = d(P1c + P2c )/dt = 0

From here I get : Fcmf = 0

But , If I sum the forces ( only weights ) of the two bodies, I don't get zero : Fcmf= -M2g -M1g ≠ 0
(I get contradiction , because I know as a fact that Fcmf must be equal to zero , Fcmf = 0 )
Why do I get a contradiction?

Homework Equations


F = ma

The Attempt at a Solution



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  • #2
CGandC said:
P1c + P2c = 0
This is true only if the CM does not accelerate, i.e. if there are no external forces acting on it. Here there is gravity.
 
  • #3
kuruman said:
This is true only if the CM does not accelerate, i.e. if there are no external forces acting on it. Here there is gravity.

so what will be the sum of momentums P1c + P2c if there are external forces acting on the system? I always get that P1c + P2c = 0 , regardless of whether there are external forces acting on the system or not.
 
  • #4
kuruman said:
This is true only if the CM does not accelerate, i.e. if there are no external forces acting on it. Here there is gravity.

This isn't correct .

OP is right that the net momentum of the two masses in CM frame is zero .

Total momentum of the system is always zero in CM frame irrespective of whether CM is accelerating or not .
 
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  • #5
CGandC said:
so what will be the sum of momentums P1c + P2c if there are external forces acting on the system? I always get that P1c + P2c = 0 , regardless of whether there are external forces acting on the system or not.

You are right :smile:

CGandC said:
the total force acting on the masses in the center of mass frame is :
Fcmf = d(P1c + P2c )/dt = 0

F = dP/dt is valid only in inertial (non accelerating) frames .

Since CM is accelerating , it is not an inertial frame .

You are right when you say that ΣP = 0 in CM frame . But you are wrong if you apply F=dP/dt in CM frame .

Note that F=dP/dt is nothing but Newton's 2nd Law which is valid only in inertial frames .

Hope that helps .
 
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  • #6
conscience said:
This isn't correct .
You are correct. I meant to say what you said much more eloquently in post #5.
 
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  • #7
conscience said:
You are right :smile:
F = dP/dt is valid only in inertial (non accelerating) frames .

Since CM is accelerating , it is not an inertial frame .

You are right when you say that ΣP = 0 in CM frame . But you are wrong if you apply F=dP/dt in CM frame .

Note that F=dP/dt is nothing but Newton's 2nd Law which is valid only in inertial frames .

Hope that helps .
Just to make sure,
If I were to use Newton's second law for the CM accelerating frame , I'd have to fix it like this ( written below ) ? :
F+Ffict=dP/dt , where Ffict is a pseudo-force / fictional force
and This pseudo force is equal to : Ffict = +M1g +M2g , so that F+Fict = 0 , Is this correct?
 
  • #8
CGandC said:
Just to make sure,
If I were to use Newton's second law for the CM accelerating frame , I'd have to fix it like this ( written below ) ? :
F+Ffict=dP/dt , where Ffict is a pseudo-force / fictional force
and This pseudo force is equal to : Ffict = +M1g +M2g , so that F+Fict = 0 , Is this correct?

I am not sure if you are properly applying the fictitious force .

This is what is happening in your example .

Fictitious force acts on all the particles whose motion we are interested in as seen from the accelerated frame .

In your example , fictitious force acts on both ##M_1## and ##M_2## in the direction opposite to the acceleration of the frame (CM frame in your example) .

So , considering downward positive , as seen from the CM frame , fictitious force ##-M_1a_{cm}## acts on ##M_1## apart from ##M_1g## .Similarly fictitious force ##-M_2a_{cm}## acts on ##M_2## apart from ##M_2g## .

##a_{cm}=g##

The net force acting on ##M_1##and ##M_2## as seen from the CM frame = ##-M_1a_{cm} - M_2a_{cm} +M_1g+M_2g ## . This evaluates to 0 .
 
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FAQ: Force in Center of Mass frame equaling zero

1. What is the center of mass frame and why is it important?

The center of mass frame is a reference frame in which the total momentum of a system is equal to zero. This frame is important because it simplifies the analysis of the motion of a system, making it easier to understand and calculate.

2. How is the center of mass frame related to the concept of force?

In the center of mass frame, the net force acting on a system is equal to zero. This means that there is no acceleration or change in velocity of the system. It is important to note that this does not mean there are no forces acting on the system, but rather that the forces are balanced and cancel each other out.

3. What does it mean when the force in center of mass frame equals zero?

This means that there is no overall force acting on the system and therefore it is in a state of equilibrium. The center of mass of the system will remain at rest or in a constant state of motion unless acted on by an external force.

4. Can the center of mass frame be applied to all systems?

Yes, the concept of the center of mass frame can be applied to all systems, regardless of size or complexity. It is a useful tool in understanding the motion of a system as a whole, rather than focusing on individual components.

5. How can the center of mass frame be used in real-world applications?

The center of mass frame is commonly used in various fields such as engineering, physics, and astronomy. It can be used to analyze the motion of objects in space, design structures that can withstand forces, and understand the stability of vehicles and aircrafts.

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