How Fast Can a Child Swing Without Breaking the Rope?

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The discussion revolves around two physics problems involving circular motion. The first problem examines the maximum speed a child can swing on a 4-meter rope before the tension doubles their weight, with the final solution suggesting a formula of v=sqrt(g/L). However, there is skepticism about the formula's validity regarding units. The second problem involves a motorcycle weighing 180 kg crossing a circular bridge with a radius of 20 m at 36 km/h, resulting in a calculated force of 865.8 N on the bridge, prompting further inquiry into the forces acting on the motorcycle during the motion. Both problems emphasize the importance of understanding forces in circular motion.
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I got this 2 problems, and final solutions.
What I need is the build up, or in other words: EVERYTHING!

please HELP!4.17.
The child swings on a rope that is 4 m (meters) long. If the tension of the rope increases to the amount that is double the weight of the child, the rope will break. What is the highest speed with which the child can pass through the lowest position of the orbit(path/trajectory)?
(Final solution: v=\sqrt{\frac{g}{L}})4.28.
Motorcycle with the driver, total weighing 180 kg, is crossing the circle-shaped arched bridge, with radius of 20 m, at average speed 36 km/h.
Calculate the force that wheels of motorcycle are pressing on the bridge while going over it?
(sorry for bad english at this one, but you'll get it)

(Final solution: F= 865,8 N)
 
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ELE55AR said:
4.17.
The child swings on a rope that is 4 m (meters) long. If the tension of the rope increases to the amount that is double the weight of the child, the rope will break. What is the highest speed with which the child can pass through the lowest position of the orbit(path/trajectory)?



(Final solution: v=\sqrt{\frac{g}{L}})

I don't think this is correct, since sqrt(g/L) does not give units of m/s.

But if the child is moving in a circular orbit, what forces are acting at the lowest point? What do the resultant of these two give?


ELE55AR said:
4.28.
Motorcycle with the driver, total weighing 180 kg, is crossing the circle-shaped arched bridge, with radius of 20 m, at average speed 36 km/h.
Calculate the force that wheels of motorcycle are pressing on the bridge while going over it?
(sorry for bad english at this one, but you'll get it)

(Final solution: F= 865,8 N)

Once again, what are the forces acting on the motorcycle as it is going over the circular arch?

(Remember your formulas from circular motion for the two problems)
 
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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