Circular Motion Loop question

[squirrelschaser]

Homework Statement

Bob starts at rest from the top of a frictionless ramp. At the bottom of the ramp, he enters a frictionless circular loop. The total mass of the child and the cart he sits in his m. What must the height of the ramp be in order for the cart to successfully traverse the loop.

r = radius of loop
h = height of ramp
theta = angle of the ramp (irrelevant though)

Homework Equations

The Attempt at a Solution

[/B]
I solved for the minimum speed at the top of the loop.

Fy = F + mg = mv^2/r

v= sqrt(rg)

I then used conservation of energy.

Initial : mgh
Final : mg2r + (m(sqrt(rg))^2)/2

mgh = mg2r + mrg/2

mgh = 5mgr/2

Cancel stuff out h = 5r/2 (WRONG)

Instead the solution calls for using kinematics not energy conservation.

v= sqrt(rg) stills hold.

vf^2 = vi^2 + 2ax

rg = 0 + 2gsin(theta)*(h/sin(theta)

rg = 2gh

r = 2h

h = r/2 (CORRECT answer)

I understand the mathematical process of the correct solution.
However, I don't understand why I can't use conservation of energy(gives me wrong answer) instead of kinematics.

 

[haruspex]

The second answer (r/2) is clearly wrong since it would not provide enough energy to reach the top of the loop even with no remaining KE.
The calculation goes wrong because it equates the speed at the bottom of the ramp to that required at the top of the loop.
The first answer is correct.

 

[squirrelschaser]

The solutions I have showed the second answer as the correct answer.

The second answer (r/2) is clearly wrong since it would not provide enough energy to reach the top of the loop even with no remaining KE.
The calculation goes wrong because it equates the speed at the bottom of the ramp to that required at the top of the loop.
The first answer is correct.

Really? That's the solution provided to me.
Glad to know I wasn't paranoid or something.

Is there any additional information that would make solving this question using kinematic possible, then?

 

[haruspex]

Is there any additional information that would make solving this question using kinematic possible, then?

The v²-u²=2as equation is effectively KE+PE constant. All that's different is factoring out the mass.

 

[HalfLostAndSearching]

I would say that both approaches are valid in solving this problem. The difference lies in the assumptions made and the equations used in each approach.

When using conservation of energy, we assume that the total energy (kinetic + potential) of the system remains constant throughout the motion. This means that we are neglecting any external forces, such as friction or air resistance, that may affect the motion. In this case, we are assuming that the ramp and the loop are both frictionless, which is not always the case in real-world scenarios.

On the other hand, when using kinematics, we are taking into account the acceleration due to gravity and the angle of the ramp. This approach is more accurate as it considers the actual forces acting on the cart as it moves through the loop.

In conclusion, both approaches can be used to solve the problem, but the results may differ due to the assumptions and simplifications made in each method. As a scientist, it is important to carefully consider the assumptions and limitations of each approach in order to choose the most appropriate one for a given problem.