How Does Increasing the Mass of the Topmost Particle Affect the Center of Mass?

[brendan3eb]

Homework Statement

What are (a) the x coordinate and (b) the y coordinate of the center of mass of the three-particle system shown in Fig. 9-22? (c) What happens to the center of mass as the mass of the topmost particle is gradually increased?

Fig 9-22 shows a particle at the origin, a particle at (1,2) and a particle at (2,1)

Homework Equations

x(com)=(x1m1+x2m2+xnmn)/M

The Attempt at a Solution

I solved parts a and b easily. And I reasoned that the center of mass should move towards the topmost particle as the mass of the topmost particle is gradually increased because the center of mass of a system of bodies by definition is the point that moves as although all of the mass were concentrated there, and thus the center of mass will be located close to the particle with the greatest mass.

I am not sure if my answer to C is sufficient. It seems too long; I would expect a more concise explanation. I also feel as though my teacher may take my argument to be circular, perhaps the question wants you to prove the definition of center of mass than use it as proof.

What do you guys and gals think?

 

[Mephisto]

look closely at your equation for the center of mass... Try to decompose the fraction and rewrite it in a different way. You could then get your conclusion mathematically just by reasoning about the proportions of the masses, and thinking about how the y(com) and x(com) change as, say, m3 starts to increase. For that matter, think also about what happens as m3 goes to infinity. (not sure if you did this in class yet, but it would show your result very nicely as well)

 

[ogb p]

I would like to provide a more detailed and concise explanation for part c. As the mass of the topmost particle is gradually increased, the center of mass will shift towards that particle. This is because the center of mass of a system is directly influenced by the mass and position of each individual particle. As the topmost particle gains more mass, it becomes a larger contributor to the overall mass of the system. This, in turn, affects the calculation of the center of mass, causing it to move closer to the particle with the greatest mass. This can also be seen in the equation for calculating the center of mass, where the mass of each particle is multiplied by its respective position. Therefore, as the mass of the topmost particle increases, the overall position of the center of mass will also shift towards that particle. This is a direct result of the definition of center of mass, which states that it is the point where the entire mass of a system can be considered to be concentrated. Hence, as the mass of the topmost particle increases, the center of mass will move towards it.