Finding the center of mass by integration.

AI Thread Summary
To find the center of mass of a uniform semicircular disk of radius R, the solution involves using integration. The approach suggested is to divide the semicircle into thin slices of thickness dz parallel to the base. By calculating the mass of each slice and integrating over dz, one can determine the center of mass. The final result shows that the center of mass is located at a distance of 4R/(3π) from the center of the circle. This method effectively applies integration to solve the problem.
tfmfyn
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Homework Statement


Show that the center of mass of a uniform semicircular disk of radius R is at a point 4R/(3(pi)) from the center of the circle.

Homework Equations


Total mass Center of mass = M rcm = m1r1 + m2r2 + ...

The Attempt at a Solution


I do not know how to apply integration to this problem to find the center of mass. Help, please?
 
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Don't you have any integral expressions for center of mass?
 
Hi tfmfyn! :smile:

Hint: divide the semicirce into slices of thickness dz parallel to the base, find the mass of each slice, and integrate … something … over dz. :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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