Easy but hard problem (loop-the-loop) normal force?

[daivinhtran]

Homework Statement

The Gravitron single-car roller coaster consists of a single loop-the-loop. The car is initially pushed, giving it just the right mechanical energy so the riders on the coaster will feel "weightless" when they pass through the top of the circular arc. How heavy will they feel when they pass through the bottom of the arc (that is, what is the normal force pressing up on them when they are at the bottom of the loop)? Express the answer as a multiple of mg (their actual weight). Assume any effects of friction or of air resistance are negligible.

Homework Equations

Fc= M(ac)
N + Mg = M(ac) ( at the top)
N - Mg = M(ac) (at the bottom)

The Attempt at a Solution

N + Mg = M(ac)
because the riders feel weightless, the normal force is zero
==>>> g = ac

So at the botoom
N - Mg = M(ac)
N -Mg = Mg
N = 2Mg <==== it's a wrong answer :(

(however the problem is in Work-energy chapters, so I think it has to deal with that)
I solved it without any energy involved

 

[SammyS]

Homework Statement

The Gravitron single-car roller coaster consists of a single loop-the-loop. The car is initially pushed, giving it just the right mechanical energy so the riders on the coaster will feel "weightless" when they pass through the top of the circular arc. How heavy will they feel when they pass through the bottom of the arc (that is, what is the normal force pressing up on them when they are at the bottom of the loop)? Express the answer as a multiple of mg (their actual weight). Assume any effects of friction or of air resistance are negligible.

Homework Equations

Fc= M(ac)
N + Mg = M(ac) ( at the top)
N - Mg = M(ac) (at the bottom)

The Attempt at a Solution

N + Mg = M(ac)
because the riders feel weightless, the normal force is zero
==>>> g = ac

So at the bottom
N - Mg = M(ac)
N -Mg = Mg
N = 2Mg <==== it's a wrong answer :(

(however the problem is in Work-energy chapters, so I think it has to deal with that)
I solved it without any energy involved

It's correct that the centripetal acceleration, a_c, is equal to g (and is downward) at the top of the loop. However, a_c does not have the same value at the bottom of the loop, because the roller coaster car is moving much faster at the bottom.