Proving COM Moves 0.5(x_1 + x_2) in Δt

[member 731016]

Homework Statement

As shown in Figure 9.22a, two blocks are at rest on a frictionless, level table. Both
blocks have the same mass m, and they are connected by a spring of negligible mass.
The separation distance of the blocks when the spring is relaxed is L. During a time
interval delta t, a constant force of magnitude F is applied horizontally to the left block,
moving it through a distance x_1 as shown in Figure 9.22b. During this time interval, the right block moves through a distance x_2. At the end of this time interval, the force F is removed.

Find the resulting speed of the center of mass of the system

Relevant Equations

Impulse–momentum theorem

For this problem,

How can we prove that the COM moves a distance 0.5(x_1 + x_2) during the time interval delta t?

Many thanks!

 

[haruspex]

Homework Statement:: As shown in Figure 9.22a, two blocks are at rest on a frictionless, level table. Both
blocks have the same mass m, and they are connected by a spring of negligible mass.
The separation distance of the blocks when the spring is relaxed is L. During a time
interval delta t, a constant force of magnitude F is applied horizontally to the left block,
moving it through a distance x_1 as shown in Figure 9.22b. During this time interval, the right block moves through a distance x_2. At the end of this time interval, the force F is removed.

Find the resulting speed of the center of mass of the system
Relevant Equations:: Impulse–momentum theorem

For this problem,

View attachment 318100
How can we prove that the COM moves a distance 0.5(x_1 + x_2) during the time interval delta t?

Many thanks!

How do you calculate the centre of mass of this system?

 

[member 731016]

How do you calculate the centre of mass of this system?

You use the COM formula @haruspex . Do you what me to calculate the initial COM or finial COM?

I found the initial COM to be,

However, the finial COM is tricky. How would you find it?

I'm think I'm meant to calculate it with respect to mass 1 again, correct? NOT with respect to where mass 1 was initially, correct?

Many thanks!

 

[haruspex]

the finial COM is tricky. How would you find it?

"left block, moving it through a distance x_1… , the right block moves through a distance x_2"

 

[member 731016]

"left block, moving it through a distance x_1… , the right block moves through a distance x_2"

Thanks, but with respect to what? The origin, or using mass 1 as the origin?

Many thanks!

 

[member 731016]

"left block, moving it through a distance x_1… , the right block moves through a distance x_2"

@haruspex, I decided to find the finial COM of the system with respect to the same place which I used to find the COM before the force was exert onto the system.

I now understand where they got their formula from. It is the finial COM - initial COM.Many thanks!