What is the minimum height for the track to make the ball go around the loop?

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In summary, the conversation discusses finding the minimum height that a track must be for a ball with a mass of .06722kg and diameter of 2.53cm to go around a loop with a diameter/height of .3m. The solution involves using conservation of mechanical energy and kinematics equations to find the initial velocity and acceleration of the ball. The conversation also addresses the issue of solving for friction and concludes with a possible solution of the minimum height being 37.5cm.
  • #1
needhlpcalc
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Homework Statement


a ball with:

mass- .06722kg
diameter- 2.53cm

rolling down a track of length 1.26m.

Find a) the minimum height the track must be for the ball to go around the loop. Loop has a diameter/height of .3m

Like this:
1m40209jicopy2.jpg

The Attempt at a Solution


I did conservation of mechanical energy with just the loop by doing KE total(bottom) = PE(top) to find the initial velocity of the ball before going up loop, giving me a tangential velocity of 1.879 m/s. I used this number to put into the kinematics equation vf=sqrrt(vi^2 + 2ad), to give me an acceleration of 1.401 m/s^2. I then used dynamics F=ma to get a force of the ball going down the slope of .09418 N, used the weight force to get a Fwt = .6594 N, and set up a triangle to find theta (call it @ here). did sin@=.09418/.6594, and solved to get a @ of 8.21 deg. I then put that angle back into the first ramp, and used trig to get a height of .18 m, or 18 cm.

If you can follow that, I have 2 questions:

1) Did i do this problem right, or am i all wrong?

2) When I try to solve for friction, I use the conservation of ME formula, to get PE - Ff = KE...however when I plug my numbers back in, I get a bigger number on the KE side then the PE side...which leads me to think I did this problem wrong in the first place? I've been working on this for about 1.5 hours now and can't seem to figure it out..past what I have done already. :( ...and yes I did add rotational KE to my totatl KE when I did the problem...
 
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  • #2
OK, so at the top of the loop Fg <= Fc for it to not fall.

So if we say...

Fg= Fc
mg=mv^2/r
g=v^2/r
v=[tex]\sqrt{gr}[/tex]

Once we know the velocity it needs to reach at the top of the loop...

Ek(at top of loop)=Eg(at top of ramp)
1/2mv^2 = mgh
v^2=2gh
[tex] \sqrt{gr}^2=2gh[/tex]
gr=2gh
r=2h
h=r/2
h=0.3/2
h=15cm

...I'm probably wrong though. It would make sense that the value needs to be greater than 0.3, simply because of conservation of energy. Good problem, interested in solution.

Thats how I would have attacked it though
 
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  • #3
EDIT: yea, that's why i thought i was wrong in the beginning...oh well. thanks for your attempt tho. :) I appreciate it.

...anyone else?
 
  • #4
Yeah, either way it doesn't make sense. The height has to be at least 0.3m. Hmmmm I like this.

I think I have it 1 sec.
 
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  • #5
Fg= Fc
mg=mv^2/r
g=v^2/r
v=[tex]\sqrt{gr}[/tex]

Eg(initial)= Eg(at top of ramp) + Ek
mgh1 = 2mgr + 1/2 mv^2
gh1=2gr + 1/2 v^2
gh1= 2gr +1/2 rg (subbed in earlier value of v)
h1=2r +1/2r
h1= 2.5r
h1= 2.5(0.15)
h1=37.5cm

Maybe?

The ball will have gravitational potential energy AND kinetic energy, in the first attempt, I only said it had kinetic energy. Let me know if my work is to messy and I'll latex it.

Can anyone confirm this solution?
 
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  • #6
OHHH...it would make sense that the ball would have BOTH KE and PE at the top of the loop! I never took that into consideration! I think you are right! :D
 

1. How does a ball go around a loop?

When a ball is released at the top of a loop, it has potential energy due to its height. As it moves down the loop, this potential energy is converted into kinetic energy, causing the ball to gain speed. The ball's inertia keeps it moving through the loop, and at the bottom, the ball's speed is greatest due to the conversion of potential energy into kinetic energy. This speed allows the ball to successfully complete the loop and continue on its path.

2. What factors affect a ball's ability to go around a loop?

The main factors that affect a ball's ability to go around a loop are its initial height, mass, and velocity. A higher initial height will give the ball more potential energy, while a greater mass will require more energy to move and may make it harder for the ball to complete the loop. Similarly, a higher initial velocity will give the ball more kinetic energy, making it easier for the ball to complete the loop.

3. Why does a ball sometimes fall off a loop before completing it?

If a ball does not have enough initial energy or speed, it may not have enough kinetic energy to complete the loop. This can also happen if the loop is not the right size or shape for the ball, causing it to lose contact with the track and fall off. Additionally, external forces such as friction or air resistance can slow the ball down and prevent it from completing the loop.

4. Can any object go around a loop?

In theory, any object can go around a loop as long as it has enough initial energy or speed to overcome gravity and maintain contact with the track. However, the size, shape, and mass of the object can greatly affect its ability to successfully complete the loop.

5. What are some real-life examples of a ball going around a loop?

One famous example of a ball going around a loop is a roller coaster. The loops on a roller coaster are designed to give the car enough speed to successfully complete the loop without falling off. Another example is a Hot Wheels track, where a small toy car goes around loops and other obstacles. In sports, a basketball or tennis ball going around the rim of a hoop or net can also be considered a form of a ball going around a loop.

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