What is the minimum height required for an object to make it around a loop?

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To determine the minimum height required for an object to successfully navigate a loop, the potential energy at the starting height must equal the potential energy at the top of the loop plus the kinetic energy needed to maintain motion. The relevant equations involve gravitational potential energy (U = mgh) and kinetic energy (K = mv^2/2). The condition for the object to stay on the track at the top of the loop is that the centripetal force must equal the gravitational force, leading to the equation v^2/r = g. By substituting these relationships, the minimum height can be expressed as h = (0.5v^2 + 2rg)/g. The discussion emphasizes the importance of frictionless conditions and starting from rest for accurate calculations.
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Homework Statement



An object slides down a slope (from rest) and then immediately around a loop (like a roller coaster). The object has mass m, and the height of the loop is 2r. What is the minimum height required for the object to make it around the loop? Express answer in terms of m, r, g, and v.

Homework Equations



K=mv^2/2
U=mgh

The Attempt at a Solution



Would the potential energy of the slope (U1) be equal to the potential energy of the loop (U2) + the kinetic energy, so h=mv^2/2 + mg2r? That's what I think so far, but not sure if it is right.
 
Last edited:
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dominus96 said:

The Attempt at a Solution



Would the potential energy of the slope (U1) be equal to the potential energy of the loop (U2) + the kinetic energy? That's what I think so far, but not sure if it is right.
Correct. Assuming that the mass starts from rest and fall a height h to the bottom of the loop, then rises to the top of the loop, the difference is the kinetic energy.

What is the condition at the top of the loop that results in a minimum h?

I presume one is assuming a frictionless interaction between mass and path (slope and loop), and no air resistance.
 
Last edited:
\frac{v^2}{r}=g
 
Astronuc said:
Correct. Assuming that the mass starts from rest and fall a height h to the bottom of the loop, then rises to the top of the loop, the difference is the kinetic energy.

What is the condition at the top of the loop that results in a minimum h?

I presume one is assuming a frictionless interaction between mass and path (slope and loop), and no air resistance.

Yes it is frictionless and starts at rest. But what do you mean condition at the loop?
 
At the top of the loop, what is the condition or requirement with respect to mass to keep it traveling in the loop, and which satisfies a minimum energy (min height) condition with respect to the initial elevation, h?
 
I have mgh = .5mv^2 + mg(2r). So do I just solve for h? Because that would be h = (.5mv^2 + mg(2r))/mg
 
dominus96 said:
I have mgh = .5mv^2 + mg(2r). So do I just solve for h? Because that would be h = (.5mv^2 + mg(2r))/mg
See post #3.
 

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