Conservation of linear momentum & center of mass

• yoni162
In summary, in order for the center of mass frame of reference to be considered inertial, the net external forces acting on the system must be equal to 0, meaning m1a1+m2a2=0. This ensures that the center of mass does not accelerate and that all forces appear to be internal to the two-car system. In cases where external forces are present, but they cancel out in pairs, the center of mass frame is still considered inertial. This can be seen in the case of two cars accelerating with equal and opposite accelerations, where the forces between the cars cancel out. Thus, the center of mass frame is inertial and linear momentum is still conserved, although one may have to account for the action
yoni162

Homework Statement

Two cars are driving along the road at constant speeds V1, V2. At t=0 they begin to accelerate with constant acceleration a1, a2. Under what circumstances is the center of mass frame of reference inertial?

Homework Equations

Fext=0 ==> change in momentum=0

The Attempt at a Solution

It's known that if there are no external forces acting on a system, the momentum is conserved. The same applies if there are external forces present, but they cancel each other so that the net external force is 0 (correct?). Now, from the center of mass (c.o.m) frame of reference, when the momentum is conserved, the total momentum is 0. Would that apply if despite the fact that there are external forces acting on the cars, if with their acceleration, the velocity of the c.o.m is kept the same (thus making it inertial)? Or is Fext=0 necessary in order for that to happen?

You are on the right track, so I will help you see through this. Suppose you are on a skateboard moving along at the velocity of the center of mass before the cars start accelerating. Now at t = 0 they start accelerating. How must the accelerations of the cars be related for you to say that Newton's Laws are obeyed and you need no fictitious external force to explain what you see?

kuruman said:
You are on the right track, so I will help you see through this. Suppose you are on a skateboard moving along at the velocity of the center of mass before the cars start accelerating. Now at t = 0 they start accelerating. How must the accelerations of the cars be related for you to say that Newton's Laws are obeyed and you need no fictitious external force to explain what you see?

OK I don't know if that's what you were going for, but I said: in order for the c.o.m frame of reference to remain inertial, we need to have the the net external forces=0.
I know that the net external forces is the velocity of the c.o.m divided by the sum of the masses, and I want it to be 0, meaning (in a one-dimensional case):

m1a1+m2a2=0

Is that true?

Also, how do I explain the fact that the original frame of reference is inertial to begin with, considering there's friction acting on the cars? Or do I say that because V is constant (for each car), the net force is 0 so there's really no problem here, even though the forces are external?

yoni162 said:
I know that the net external forces is the velocity of the c.o.m divided by the sum of the masses, and I want it to be 0, meaning (in a one-dimensional case):

m1a1+m2a2=0

Is that true?

It is. If the above condition is satisfied, the center of mass does not accelerate, i.e. it is an inertial frame. Furthermore, an observer in that frame will interpret the above condition to mean that the two cars exert equal and opposite forces on each other in accordance with Newton's 3rd Law, i.e. all forces appear to be internal to the two-car system.

Also, how do I explain the fact that the original frame of reference is inertial to begin with, considering there's friction acting on the cars? Or do I say that because V is constant (for each car), the net force is 0 so there's really no problem here, even though the forces are external?

Correct. A frame is inertial as long as it is not accelerating. When both cars are moving at constant velocity, their center of mass is not accelerating.

One more thing I wanted to ask..is there conservation of linear momentum in this case? Since the forces acting on the cars aren't internal forces after all..

This is a philosophical question. If you define, strictly, that momentum is conserved as long as the center of mass does not accelerate, yes, linear momentum is conserved. The non-conservation of linear momentum when external forces act on the system comes about when internal forces act in pairs. Here, if m1a1+m2a2=0, internal forces cancel in pairs. Therefore, by the strict definition, linear momentum is conserved. An observer in the CM frame will not be able to tell otherwise although he/she may have to account in some way or another for the action-reaction apparent force between the cars. In accelerating frames, one has to invent fictitious external forces; this is a special case where one has to invent a fictitious internal force.

1. What is the law of conservation of linear momentum?

The law of conservation of linear momentum states that the total momentum of a system of objects remains constant unless an external force is applied to the system. This means that in a closed system, the total momentum before a collision or interaction will be equal to the total momentum after the collision or interaction.

2. How is the conservation of linear momentum related to Newton's Third Law?

Newton's Third Law states that for every action, there is an equal and opposite reaction. This means that when two objects interact, the forces they exert on each other will be equal and opposite. This also means that the total momentum of the two objects will be conserved, as there is no net external force acting on the system.

3. What is the center of mass and how is it related to conservation of linear momentum?

The center of mass is the point in a system where the mass is evenly distributed in all directions. In a closed system, the center of mass will remain at a constant velocity unless acted upon by an external force. This is related to conservation of linear momentum as the total momentum of the system is equal to the product of the total mass and velocity of the center of mass.

4. Can the conservation of linear momentum be violated?

No, the conservation of linear momentum is a fundamental law of physics and has been rigorously tested and proven to hold true in all scenarios. However, in some cases, it may appear that momentum is not conserved due to external forces that are not easily observable or accounted for.

5. How is the conservation of linear momentum applied in practical situations?

The conservation of linear momentum is applied in various practical situations, such as in collisions between objects, rocket propulsion, and sports movements. It is also used in designing structures and vehicles to ensure stability and balance. Additionally, the concept of center of mass is used in designing structures like bridges and buildings to distribute weight evenly and prevent collapse.

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