A loop-the-loop track problem

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In summary, the problem is to find the minimum initial height h above the bottom of the track that a small solid marble of mass m and radius r must be released from rest in order to roll without slipping along a loop-the-loop track with radius R. Using circular motion and conservation of energy, the necessary speed at the top of the loop is found to be V=Sqr(g(R-r)). This leads to the minimum initial height h=2.7(R-r) that is needed to achieve the required speed.
  • #1
RIT_Rich
A small solid marble of mass m and radius r will roll without slipping along the loop-the-loop track shown in Fig. 12-34 if it is released from rest somewhere on the straight section of track. For the following answers use m for the mass, r for the radius of the marble, R for the radius of the loop-the-loop and g for the acceleration due to gravity.

Here's the picture:

12_34.gif


a) a) From what minimum height h above the bottom of the track must the marble be released to ensure that it does not leave the track at the top of the loop? (The radius of the loop-the-loop is R. Assume R>>r.)

Can anyone give me any hints as to how to do this problem, becasue to tell you the truth I don't even know how to start it.

Thanks a lot
 

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  • #2
At first I thought this was a "what speed do we have to have in order that centrifugal force will be equal to gravitational force and keep the marble from falling off the track" but you aren't ASKED for speed. You are simply asked for initial height. Since you didn't give any "friction" terms, I'll take it you are ignoring friction and "conservation of energy applies". You simply need to make sure the marble still has SOME speed at the top and that should make the minimum initial height obvious!

(Another hint: If you have a pendulum with no friction and release the bob 1 meter above it's lowest point, how high will it rise on the other side?)
 
  • #3
I don't think so. HallsofIvy's first thought was better.

You aren't asked for speed, but you are asked for the height that will result in the necessary speed.

First, you need an expression for the speed that's needed at the top of the loop such that the force provided by gravity is exactly equal to the centripetal force needed to keep the ball in circular motion (so the ball doesn't fall). For this, use circular motion/Newton's 2nd Law.

THEN, you can use conservation of energy to find the minimum initial height that will achieve the required speed at the top of the loop.
 
  • #4
Right, first I needed to find the speed that it has to have at the top of the loop, which was V=Sqr(g(R-r)). Than using conservation of energy I got H to be 2.7 (R-r)...which was the right answer.

Thanks a lot everyone.
 

1. What is a loop-the-loop track problem?

A loop-the-loop track problem is a physics problem that involves a small object (such as a toy car) moving around a circular track and encountering a loop, or vertical circle. The challenge is to determine the minimum speed required for the object to successfully complete the loop without falling off the track.

2. What is the significance of a loop-the-loop track problem?

Loop-the-loop track problems are often used as a real-life application of the laws of physics, specifically the principles of centripetal force and conservation of energy. They also provide an interesting and engaging way to practice problem-solving and critical thinking skills.

3. How do you solve a loop-the-loop track problem?

The key to solving a loop-the-loop track problem is to set up and apply the equations for centripetal force and conservation of energy. These equations involve variables such as mass, velocity, radius of the loop, and acceleration due to gravity. By rearranging the equations and plugging in known values, you can solve for the minimum speed required for the object to complete the loop.

4. What factors affect the solution to a loop-the-loop track problem?

The solution to a loop-the-loop track problem is affected by several factors, including the mass of the object, the radius of the loop, and the acceleration due to gravity. The solution will also vary depending on whether the track is frictionless or not, and whether the object is starting from rest or moving with an initial speed.

5. Are there any real-life examples of loop-the-loop track problems?

Yes, there are several real-life examples of loop-the-loop track problems. Roller coasters, for example, often incorporate vertical loops into their design. Also, some amusement park rides, such as the "Gravitron," use centrifugal force to keep riders pinned against the walls as the ride spins around a circular track.

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