Loop-the-loop marble physics

In summary, to find the minimum height h required for a marble to make it around a loop-the-loop without falling off, you will need to use a combination of energy and Newton's second law. By finding the velocity of the marble as a function of its height, you can determine the starting height that will give the marble enough velocity at the top of the loop for it to continue in a circular path without the need for a normal force from the track. This minimum height will prevent the marble from falling off the track.
  • #1
dmahmoudi
3
0
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?

I'm stumped - I'm assuming I need to use energy for this problem, but how do I factor in the loop-the-loop. Maybe with rotational kinetic energy?
 
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  • #2
dmahmoudi said:
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?

I'm stumped - I'm assuming I need to use energy for this problem, but how do I factor in the loop-the-loop. Maybe with rotational kinetic energy?

There are two forces on the marble - the normal force applied by the track and the force of gravity. At the top of the track both forces point down towards the centre of the track. By Newton's second, the sum of these forces cause a centripetal acceleration. Use energy to find v and thus the centripetal acceleration at the top of the loop. Use Newton's second to find the normal force at the top. Find the height at which the normal force is zero. If released above this height, the marble goes loop-the-loop. If released below this height, the marble goes plop.

Regards,
George
 
  • #3
dmahmoudi said:
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?

I'm stumped - I'm assuming I need to use energy for this problem, but how do I factor in the loop-the-loop. Maybe with rotational kinetic energy?

You will need both rotational and translational kinetic energy to find the velocity of the marble as a function of its height relative to the starting height. You will then need to find the starting height that will give the marble the velocity at the top of the loop such that gravity will be just sufficient to provide the centripetal force needed for the circular path of the marble (i.e., there will be no normal force from the track at the top of the loop). Any lower velocity at the top would result in the marble being separated from the track by the gravitational force (separation would occur before the marble reached the top of the loop).
 
  • #4
Awesome, so I see. Thanks for the help guys!
 

1. How does a loop-the-loop marble track work?

A loop-the-loop marble track works by utilizing the principles of centripetal force and gravity. As the marble travels through the loop, it experiences an upward force from the track which keeps it moving in a circular path. At the top of the loop, the marble's velocity is directed downwards, but the centripetal force keeps it moving in a loop rather than falling off the track.

2. What factors affect the success of a loop-the-loop marble track?

The success of a loop-the-loop marble track depends on several factors, including the height and radius of the loop, the speed of the marble, and the shape and smoothness of the track. These factors affect the amount of centripetal force and the friction between the marble and the track, which ultimately determines whether the marble will complete the loop or fall off.

3. Can any object be used as a marble for a loop-the-loop track?

In theory, any object can be used as a marble for a loop-the-loop track as long as it has enough mass to create sufficient centripetal force and can roll smoothly along the track. However, small, round objects with a smooth surface are often the best choice for optimal performance.

4. How do you calculate the minimum speed needed for a marble to complete a loop-the-loop track?

The minimum speed needed for a marble to complete a loop-the-loop track can be calculated using the formula v = √(gr), where v is the minimum speed in m/s, g is the acceleration due to gravity (9.8 m/s^2), and r is the radius of the loop in meters. This formula assumes a perfect circular loop and neglects frictional forces.

5. Are there any real-life applications for loop-the-loop marble physics?

Loop-the-loop marble physics has several real-life applications, including roller coasters and other amusement park rides, as well as in engineering and design of circular structures such as bridges or tunnels. It also provides a fun and interactive way to learn about physics and principles of motion.

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