Proving COM Moves 0.5(x_1 + x_2) in Δt

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The discussion revolves around proving that the center of mass (COM) of a system moves a distance of 0.5(x_1 + x_2) during a time interval Δt when two blocks connected by a spring are acted upon by a force. Participants clarify the process of calculating the initial and final positions of the COM, emphasizing the importance of consistency in reference points. The initial COM is established, but determining the final COM proves more complex, leading to discussions about using the same reference for both calculations. Ultimately, the participants conclude that the formula for the distance moved by the COM is derived from the difference between the final and initial COM positions. The conversation highlights the application of the impulse-momentum theorem in this context.
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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,

1670034935698.png

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

Many thanks!
 
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Callumnc1 said:
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?
 
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haruspex said:
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,
1670035936964.png

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!
 
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Callumnc1 said:
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"
 
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haruspex said:
"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!
 
haruspex said:
"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.
1670040402651.png


I now understand where they got their formula from. It is the finial COM - initial COM.Many thanks!
 
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Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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