Roller Coaster Car Normal Force: 3693 N and -1307 N at Various Velocities

[Theorγ]

Homework Statement

A roller coaster car has a mass of 1200 kg when loaded with passengers. As the car passes over the top of the circular hill of radius 18 m, its speed is not changing. What are the magnitude of the normal force and the direction of that normal force with the below conditions?

Condition A = When velocity is 11 m/s
Condition B = When velocity is 14 m/s

Homework Equations

a_c = v²/r
F_net = ma_c

The Attempt at a Solution

I have already solved the whole problem, but I would like you guys to check if my answers are correct:

Condition A: F_n = 3693 N, Direction is Up
Condition B: F_n = -1307 N, Direction is Down

 

[Sniperman724]

For condition A, i keep getting a larger force upward, and for condition B I keep getting a much larger number than what you got, and I am not sure how you got a negative number to be honest. Try double checking that you squared the velocity and divided by the radius 18m

 

[Theorγ]

Did you write the acceleration as negative for both conditions?

 

[Sniperman724]

No I did not, I used the equations that you gave above. I squared the velocity, divided by the radius, then multiplied by the mass of the roller coaster.

 

[Theorγ]

So then should acceleration be negative? I only thought it was negative because I was under the impression that the direction of acceleration was pointing downwards once the roller coaster was at the top of the hill.

 

[gneill]

So then should acceleration be negative? I only thought it was negative because I was under the impression that the direction of acceleration was pointing downwards once the roller coaster was at the top of the hill.

The centripetal force is "supplied" by gravity when the cars are moving slowly. In terms of the net force pressing downwards on the track, the centripetal force is subtracted from the weight (If you speak in hushed tones and mumble quietly about accelerated frames of reference, you may think in terms of centrifugal force which is outward directed and countering gravity in this case)

As the speed increases, so does the centripetal force requirement. Once the available "allotment" of gravitational force is "used up" the cars no longer press down on the rails. Presumably the roller coaster has some method of holding onto the rails (and presumably the occupants are strapped in!), otherwise the roller coaster cars will separate from the rails at higher speeds.

Now, where does this leave us in terms of normal force? If you calculate the total force of the cars downwards then the normal force, being the reaction force of the ground due to the applied force, is simply the negative of that value. The downward directed weight is mg and the centripetal (mumble mumble centrifugal mumble) force subtracts from it, so

So, Fn = -(mg - Fc) .

 

[Theorγ]

That means the answers I calculated are reversed? Condition A should be negative and Condition B should be positive?

 

[gneill]

That means the answers I calculated are reversed? Condition A should be negative and Condition B should be positive?

No. Remember, the normal force (reaction force) is the opposite of the net "weight". You did fine.