Hard/Tricky Problem on Energy Conservation

AI Thread Summary
To determine the largest radius R for a car moving at an initial speed of 4.0 m/s to maintain contact with a circular track, it's essential to apply the principles of energy conservation and centripetal force. The potential energy (PE) and kinetic energy (KE) equations are relevant, alongside the centripetal force formula. The initial assumption that the car has zero velocity at the top of the circle is incorrect, as the car must maintain contact throughout its motion. The problem is sourced from "Physics 6th Edition" by Cutnell & Johnson, which does not provide an answer in the back. Clarification on the attachment's visibility was also requested.
ShamTheCandle
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Homework Statement


If the car is given an initial speed of 4.0 m/s, what is the largest value that the radius R can have if the car is to remain in contact with the circular tract at all times?
See the attached file (named: car.pdf) for the drawing.

Homework Equations


PE=mgh ----> Formula for Potential Energy
KE=(1/2)mv² ----> Formula for Kinetic Energy
Fc=mv²/R ----> Formula for Centripetal Force

The Attempt at a Solution


My initial attempt was to assume that the car has zero velocity at the top of the circle. However this is not true, since the car should "remain in contact with the circular tract at all times". This is an even problem from my textbook, so no answer is given in the back. By the way, I am referring to "Physics 6th Edition" by Cutnell & Johnson.

Thank you for helping!
 
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I am sorry if the attachment didn't show up. Here is it.
 

Attachments

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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