What Height Is Needed for a Ball to Complete a Loop-the-Loop?

In summary, the ball must be released at a height of h above the top of the loop in order for it to just make it around the loop. The ball has a minimum speed at the top of the loop in order to maintain contact.
  • #1
huskydc
78
0
A small ball of radius r = 2.4 cm rolls without slipping down into a loop-the-loop of radius R = 2.5 m. The ball has mass M = 352 gm.

How high above the top of the loop must it be released in order that the ball just makes it around the loop?
 
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  • #2
Hint: What minimum speed must the ball have at the top of the loop in order for it to maintain contact? (It's not zero!)
 
  • #3
sorry doc, i really have no clue on this...i know i need to find the speed. since here...the ball rolls w/o slipping. thus the velocity is equal to its tangential velocity.
 
  • #4
Consider the forces on the ball at the moment it reaches the top of the loop. Apply Newton's 2nd law, realizing that the ball is undergoing circular motion.
 
  • #5
The ball will start falling if the Normal force exerts to it is 0 N.
Viet Dao,
 
  • #6
ok, i got it, thanks!
 
Last edited:
  • #7
huskydc said:
A small ball of radius r = 2.4 cm rolls without slipping down into a loop-the-loop of radius R = 2.5 m. The ball has mass M = 352 gm.

How high above the top of the loop must it be released in order that the ball just makes it around the loop?

You do not need to calculate actual forces, you know. Just apply conservation of total energy by splitting up the motion in two parts.
1) begin : the height you start from (h)
2) end : the point where you start the loop (height = 0)

and
1) point where you start the loop (same as 2) above)
2) point at the top of the loop (indeed the normal force is just zero here)

For part one you have (R is the radius of the loop)
[tex]mgh = \frac{mv^2}{2}[/tex] : this gives v

For part two : at the top of the loop, you have [tex]\frac{mv^2}{R} = mg[/tex] : from this you have v' at the top of the loop.

Applying energy conservation for part two yields : [tex]\frac{mv^2}{2} = mg2R + \frac{mv'^2}{2}[/tex]. If you replace the v and v' by their calculated expressions you will get [tex]h = \frac{5R}{2}[/tex]

Remember that this only counts for the ball being a point particle. I am not incorporating rotation here. If you wanna, the total kinetic energy must be expanded with the rotational kinetic energy [itex]1/2I\omega^2[/itex]

regards
marlon
 
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1. How does a ball roll in a loop?

When a ball is placed at the top of a loop, it has potential energy due to its position. As it begins to roll down the loop, this potential energy is converted into kinetic energy, causing the ball to gain speed. This speed allows the ball to maintain enough momentum to travel through the loop and back to the starting point.

2. What factors affect a ball's ability to roll in a loop?

There are several factors that can affect a ball's ability to roll in a loop, including the size and shape of the loop, the mass and size of the ball, and the amount of friction present. The steeper the loop, the more potential energy the ball will have at the top, allowing it to travel through the loop more easily. A heavier ball may also have more momentum to overcome any friction present.

3. Can a ball roll in a loop without any external force?

No, a ball cannot roll in a loop without any external force. As mentioned before, the ball relies on its initial potential energy and the conversion to kinetic energy to maintain its momentum and complete the loop. Without this initial push, the ball would simply fall to the ground.

4. Why does a ball sometimes fall out of the loop?

A ball may fall out of a loop if it does not have enough speed or momentum to overcome the force of gravity. This can happen if the loop is too flat or if there is too much friction present, causing the ball to lose its momentum and fall out of the loop. Additionally, if the ball is not placed at the exact height and angle required, it may not have enough potential energy to complete the loop.

5. What are some real-world applications of ball rolling in a loop?

Ball rolling in a loop is a common concept used in amusement park rides, such as roller coasters and water slides. It is also seen in sports, such as skateboarding and BMX biking, where athletes use ramps and loops to perform tricks and stunts. Additionally, this concept is used in engineering and design when creating tracks and pathways for objects, such as conveyor belts or assembly lines.

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