Determining Minimum Height for Marble to Complete Loop-the-Loop

In summary, to find the minimum height h for a marble of mass m and radius r to make it around a loop-the-loop of radius R, we must take into account the center of mass and use the constraint of velocity at the top of the loop. The final energy at the top can be calculated using the kinetic energy and rotational energy. Setting this equal to the initial energy, we can solve for h.
  • #1
kamikaze1
22
0

Homework Statement


A marble rolls down a track and around a loop-the-loop of radius R. The marble has mass m and radius r. What minimum height h must the track have for the marble to make it around the loop-the-loop without falling off?

Homework Equations



What minimum height h must the track have for the marble to make it around the loop-the-loop without falling off?

The Attempt at a Solution


Find initial energy: mgh
Final energy tabulation: mg(2R) + 0.5mrg + 0.5(2/5)mr^(2)(v/r)^2
Set equal to initial energy, I obtain 2R+7/10 =h, which isn't the right answer, what did I do wrong?

I found the velocity at top by using Newton's 2nd law, i get v = squareroot(rg). I find w using constraint of wr=v. I get 2/5(mr^2) using moment of inertia for a solid sphere.

Any help would be appreciated. I'm so confused why this doesn't work out.
 
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  • #2
I'm still new to this, but it looks like you've got everything right except for the last part of your final energy calculation (the 0.5(2/5)mr^(2)(v/r)^2 part). I'm not quite sure why it's there, but I'm pretty sure that's the part that's throwing you off.
 
  • #3
SchruteBucks said:
I'm still new to this, but it looks like you've got everything right except for the last part of your final energy calculation (the 0.5(2/5)mr^(2)(v/r)^2 part). I'm not quite sure why it's there, but I'm pretty sure that's the part that's throwing you off.

The last part is the kinetic rotational energy using I = (2/5)(M)(R^2)
Using kinetic rotational energy = 0.5Iw^2, I have
0.5(2/5)(MR^2)w.
using constraint wr=v, knowing that v=(rg)^0.5, I get
0.5(2/5)mr^(2)(v/r)^2

The answer isn't right.
 
  • #4
The energy at the top of the loop is the potential (which you have) plus it's kinetic, since it is moving in a circle there must be an acceleration at the top equal to [itex] \frac{V^2}{R} [/itex], for this to be a minimum (IE only just making it around the loop) this acceleration must be set equal to [itex]g[/itex] so we have
[tex] V^2 = gR \Rightarrow KE = \frac{1}{2}mgR [/tex]
[tex]\displaystyle{\dot{. .}} mgh = mg(2R) + \frac{1}{2}mgR [/tex]
Which you can then solve for the minimum height :)

EDIT: Just reread your question, the "effective" radius of the loop is changed because of the radius of the marble (the center of mass of the marble moves around a slightly smaller loops) and there is indeed rotational energy equal to [itex]\frac{1}{2}\frac{2}{5}\frac{gR}{r^2}mr^2 [/itex] (again this has to be changed because the [itex]R[/itex] isn't really the radius of the loop that the center of mass moves over, but I'll leave that to you)

:)
 
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  • #5
Never mind. Thanks for all your suggestions. I've just solved them. I left out the little tiny details which messed up the problems. I didn't take into account of the center of mass, which I paid dearly.
Here is how I solved it.
Initial energy: mg(h+r)
At the loop, the radius is R-r because we're taking into account of the center of mass. Hence using Newton's 2nd law.
m(v^2)/(R-r) = mg, solve for v = (g(R-r))^0.5 (1)

Final energy at top of the loop is:
mg(2R-r) + 0.5(m)(g(R-r)) + 0.5(2/5)(mr^2)(w^2).
Using velocity constraint, w=v/r=(1/r)(g(R-r))^0.5
Plug that in (1), and set (1) equal to initial energy, then solve for h.
 

FAQ: Determining Minimum Height for Marble to Complete Loop-the-Loop

What is a loop-the-loop?

A loop-the-loop is a stunt or trick performed by objects, usually marbles, on a track, where the object completes a full loop without falling off the track.

What is the minimum height required for a marble to complete a loop-the-loop?

The minimum height required for a marble to complete a loop-the-loop depends on various factors such as the size and weight of the marble, the shape and angle of the track, and the velocity of the marble. Generally, a minimum height of 2-3 times the diameter of the marble is required for it to complete a loop-the-loop successfully.

How can you determine the minimum height for a marble to complete a loop-the-loop?

The minimum height for a marble to complete a loop-the-loop can be determined by using the laws of physics, specifically the laws of motion and energy. By considering the velocity and acceleration of the marble, as well as the forces acting on it, the minimum height can be calculated using equations such as the conservation of energy and the centripetal force equation.

What materials are needed to build a loop-the-loop track for marbles?

The materials needed to build a loop-the-loop track for marbles include a sturdy base, a track made of smooth and durable material such as plastic or metal, supports to hold the track in place, and a launching mechanism or ramp to give the marbles enough velocity to complete the loop.

Are there any safety precautions to consider when building and testing a loop-the-loop track for marbles?

Yes, there are safety precautions to consider when building and testing a loop-the-loop track for marbles. The track should be sturdy and secure to prevent it from collapsing or falling during use. It is also important to wear safety gear such as goggles when testing the track to protect against flying marbles. Furthermore, make sure to test the track in a safe and controlled environment away from people or objects that may get hit by the marbles.

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