Physics Conservation of Energy Question

[beyondpaperchase]

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How would you go about answering this question?

I have looked at the key, but there are parts of it that I am confused about. For instance: At the top of the loop, we must derive another formula to connect ideas together by using Newton’s Second Law to describe the forces acting on the cart and then to solve for v: N - mgcosθ = mv^2/r --> rgcosθ = v^2 --> v = (gr)^.5

I am assuming N refers to normal force, but what does the "-mgcosθ" refer to?

Then the key goes on to say: This expression is for the velocity at the top of the loop since there is no normal force only at that point. Additionally, h = 2r such that the potential energy is mg(2r).
How did we come to the realization that h=2r?

Any clarifications/explanations would be greatly appreciated. Thanks in advance.

 

[aldol16]

Alright, so let's reason this out. At the top of the loop, the only thing providing you with any centripetal force is the force due to gravity because there's no normal force at the top of the loop. Centripetal force is given by m*v^2/r. Therefore, m*g = m*v^2/r. Simplifying that gives you v = sqrt(g*r). That's your velocity at the top of the loop.

Conservation of energy can also be applied such that the sum of your kinetic energy and potential energy at the top of the loop must be equal to the potential energy that you had at the top of the hill. That's m*g*h = 1/2*m*v^2 + m*g*2*r. Simplifying, m*g*(h-2*r) = 1/2*m*v^2 and g*(h-2*r) = 1/2*v^2. Substituting in from above, g*(h-2*r) = 1/2*g*r. Solving for h, you get h = 1/2*r+2*r = 5/2*r. So I get 25 m.

Then the key goes on to say: This expression is for the velocity at the top of the loop since there is no normal force only at that point. Additionally, h = 2r such that the potential energy is mg(2r).
How did we come to the realization that h=2r?

I think there is some confusion of variables here. This "h" must refer to the height of the cart at the top of the loop, not at the top of the hill. This is because at the top of the loop, the cart is a circle's diameter above the ground, or 2 times the radius. This "h" is important because it gives the potential energy of the cart at the top of the loop.