Standing Wave Problem: Aluminum and Steel Wire Joint with External Source

AI Thread Summary
An aluminum wire and a steel wire are joined, creating a compound wire system with specific dimensions and a mass load. The problem involves calculating the lowest frequency of excitation for standing waves, ensuring that the joint acts as a node. Initial calculations of mass and linear density were performed, but discrepancies in wave speed and frequency between the two wire sections were noted. The user resolved their confusion shortly after posting, indicating a quick realization of the solution. The discussion highlights the complexities of wave behavior in composite materials under load.
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an aluminum wire of length L1 = 50.0 cm, cross-sectional area 1.00×10-2 cm2, and density 2.70 g/cm3, is joined to a steel wire of density 7.80 g/cm3 and the same cross-sectional area. The compound wire, loaded with a block of m = 10.0 kg, is arranged so that the distance L2 from the joint to the supporting pulley is 44.13 cm. Transverse waves are set up in the wire by using an external source of variable frequency; a node is located at the pulley. Find the lowest frequency of excitation for which standing waves are observed such that the joint in the wire is one of the nodes.

Having some difficulty with this problem. First I calculated the mass and linear density of each section of the rope. Using v^2 = T/u the wavespeed on each section was calculated. However, this gives a different frequency for each section, so I'm obviously doing something wrong. Any suggestions?
 
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haha nevermind got that one literally 1 min after posting...just came to me.
 
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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