Center of Mass Homework: Equations & Solution

AI Thread Summary
The discussion focuses on solving a homework problem related to finding the center of mass. The correct answer is identified as Y = 4R/(3*pi), but the initial approach was flawed due to not calculating the x and y components of the center of mass vector separately. Participants emphasize the importance of integrating the vector components first, as they change with angles. Acknowledgment of common mistakes in the problem-solving process is noted, reinforcing the learning experience. Understanding the correct method for integration is crucial for accurate results in center of mass calculations.
snowcover
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Homework Statement



Capture.JPG

Homework Equations


Please see attachment to #3.


The Attempt at a Solution


Attempt Center of Mass 001.jpg


Apparently, the answer is Y= 4R/(3*pi). Why?
 
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snowcover said:
Apparently, the answer is Y= 4R/(3*pi). Why?
You are asking the wrong question. The right question to ask is, "Where did I go wrong?" You went wrong in that you did not calculate the x and y components of the center of mass vector. You need to calculate the x and y components of that vector separately.
 
Thanks. I separated vector r into x and y components, integrated, and I got it.

I guess my first attempt did not work because even though I tried to do a conversion at the end...I needed to integrate r as a vector first because its components change as the angles change.
 
snowcover said:
I guess my first attempt did not work because even though I tried to do a conversion at the end...I needed to integrate r as a vector first because its components change as the angles change.
You got it. It's a common mistake to do what you did. Now you know better. :smile:
 
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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