Velocities seen from the center of mass

[dapias09]

Hello all,

I'm confused with the center of mass concept. Regard the following problem, we have a spring with negligible mass joined to two masses m1 and m2 (m1=m2) on a table without friction force. At t=0 a thrust Vo (initial velocity) is given to m2 with direction perpendicular to the spring direction (see figure adjoint). How are the velocities at t=0 seen from the center of mass?

At t=0 the m1 velocity seen from the laboratory is zero but this particle isn't fixed on the table.

Well, I guess that I should calculate the velocity of the center of mass as:
Vcm = m1V1 + m2V2 / (m1 + m2) = m2V2 / (m1 +m2) = Vo / 2, since m1 and m2 are equals.

So the initial situation is equivalent to give a thrust to the center of mass with velocity Vo/2, but how I know what are the velocities of m1 and m2 seen of the center of mass?

Can anyone help me?

Thanks in advance.

 

[James_Harford]

Hello all,

I'm confused with the center of mass concept. Regard the following problem, we have a spring with negligible mass joined to two masses m1 and m2 (m1=m2) on a table without friction force. At t=0 a thrust Vo (initial velocity) is given to m2 with direction perpendicular to the spring direction (see figure adjoint). How are the velocities at t=0 seen from the center of mass?

At t=0 the m1 velocity seen from the laboratory is zero but this particle isn't fixed on the table.

Well, I guess that I should calculate the velocity of the center of mass as:
Vcm = m1V1 + m2V2 / (m1 + m2) = m2V2 / (m1 +m2) = Vo / 2, since m1 and m2 are equals.

So the initial situation is equivalent to give a thrust to the center of mass with velocity Vo/2, but how I know what are the velocities of m1 and m2 seen of the center of mass?

Can anyone help me?

Thanks in advance.

Ask yourself, instead, what are the velocities of m1 and the center of mass seen from m2. If you can figure that out for m2, use the same method for the center of mass.

 

[dapias09]

Good idea!, James, thank you.

Just to check, isn't it -Vo for m1 and -Vo/2 for the center of mass seen from m2?
In this way, I have another question, do you know what should I do if the masses (m1 and m2) are different.

Greetings.

 

[James_Harford]

Good idea!, James, thank you.

Just to check, isn't it -Vo for m1 and -Vo/2 for the center of mass seen from m2?
In this way, I have another question, do you know what should I do if the masses (m1 and m2) are different.

Greetings.

In regards to your first question, if they are not, what does that say about the principle of (Galilean) relativity?

In regards to your second, clearly, the center of mass speed needs to be recalculated. The rest is the same as before :)

 

[PJC01]

Hello,

Firstly, let's define the center of mass as the point at which the entire mass of a system can be considered to be concentrated. In the given scenario, the system includes the spring and the two masses (m1 and m2). The center of mass would be located at the midpoint between m1 and m2, since they have equal masses.

Now, let's consider the velocities of m1 and m2 from the center of mass. At t=0, the velocity of the center of mass, Vcm, is indeed equal to Vo/2, as you have correctly calculated. This is because the total momentum of the system is conserved, and the initial momentum of the system is Vo, which is equal to the final momentum of the system (since there are no external forces acting on the system).

From the perspective of the center of mass, m1 and m2 would have opposite velocities, with m1 moving in the negative direction and m2 moving in the positive direction. This is because, as you mentioned, the spring exerts a force on both masses, causing them to move in opposite directions. However, the velocities of m1 and m2 would not be equal. In fact, the velocity of m1 would be larger than that of m2, since m1 is closer to the center of mass and therefore experiences a larger force from the spring.

I hope this helps clarify the concept of velocities seen from the center of mass. In summary, the initial velocity of the center of mass is equal to Vo/2, but the velocities of m1 and m2 from the center of mass would be unequal due to their different distances from the center of mass. Let me know if you have any further questions. Good luck with your problem-solving!