Physics!

Hey Alex,

So for questions when they have a hill leading up to a mini-roller coaster : like in the Force, Motion, Gravity and Equilibrium Test 1, the solution had said that whenever you're looking at minimum velocity (vc) Normal Force is zero, why is that?  (This is #5 by the way)

Also, one of the questions (#2) tested how friction would affect Normal Force. I know they gave it in terms of the equations of Force and Centripetal force. Is there an intuitive way of thinking about it without having to resort to looking at the equations? I understand how they got the answer, but I definitely wouldn't have thought to go derive the answer from the equations. Was that unavoidable for this question?

Thanks!
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For the first question, the general idea is that we're looking for the minimum speed at which a cart will just make it around a loop, without falling off at the top.

Think about what we do for any kinematics question:
1.  Make a Free Body Diagram
2.  Newton's 2nd Law
3.  Solve for what they want

At the top, we'll have the following forces:
-gravity.  Points straight down, with magnitude mg
-normal force.  Points inwards at any point in the circle, since it is always perpendicular to the surface of contact.  Magnitude may vary in this case.
Those are all the forces.  We can assume friction is negligible in the passage unless otherwise specified, so we don't have to worry about it here.

Newton's Second Law:  Fnet = ma.  In this case, Fnet is equal to the centripetal force, since we're going around a circle.  Centripetal force has a special equation -- Fc = mv^2/r.  So, if we're at the top and we're just making it around, the entire centripetal force should be accounted for by gravity.  This is why the normal force is zero -- at the minimum speed, the entire centripetal force is due to gravity.  At faster speeds, there will also be a normal force that contributes to the centripetal force.  Thus, having our normal force equal to zero is the set up:
Fc = mv^2/r = mg.  From there, we could solve for the minimum speed itself if they asked.  In this case, the question just asks what happens when the mass in increased.  From the equations, we know it's not important -- mass cancels out in the set up (v^2/r = g).

 For question #3 - I think that's the one you actually meant, you can do this intuitively.  What will friction do?  Friction will slow the cart down.  What happens when the cart is slowed down?  Well, the net force must be lower -- think Newton's Second Law.  Fnet = Fc = mv^2/r.  If v is lower, and r is still the same, the net force is less.

The net force is equal to the centripetal force, and the normal force always points the same direction as the centripetal force (radially inwards).  So if we know the net force (and by extension, the centripetal force) is less, then it stands to reason that the normal force is always smaller.