Rotational Inertia: Pipe Reaches Bottom First?

In summary, the object with the highest translational kinetic energy, which is the ball, will reach the bottom of the inclined plane first. This is due to the fact that when an object rolls without slipping, the kinetic energy can be separated into a translational and rotational component, and the object with the highest rotational kinetic energy will have a lower translational kinetic energy and therefore a higher speed. The height of the incline is not needed to determine the answer, only a comparison of the final velocities.
  • #1
NoHeart
28
0
if a ball, a solid cylinder, and a hollow pipe each with equal masses and radius are released simultaneously from an inclined plane, which will reach the bottom first?
i would say the pipe would reach the bottom first, because the rotational inertia of the pipe is greater than that of the other two. can anyone tell me if I'm correct? if not, how do i go about answering this question (if inertia has nothing to do with it)?
 
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  • #2
It has everything to do with principal moments of inertia.I'm assuming they roll without slipping and there's no friction on the incline.(Since the problem doesn't say so,it's better to assume it,for simplicity).So u'd have to compare the linearvelocities once they reach the ground...

Daniel.
 
  • #3
You have identified the important property of the objects, but you have come to the wrong conclusion. Think about energy conservation.
 
  • #4
inert

yeah, I'm totally backwards. so the object with the least amount of inertia would have the greatest velocity, that would be the cylinder...but you are saying it has to do with energy conservation- why?
using potential energy=kinetic energy, the ball has the fastest speed. I'm going to say that's the answer, but why?
 
  • #5
When an object rolls without slipping, all the little pieces of mass are moving at different speeds, which makes calculating kinetic energy different than when all the pieces are moving at the same speed. Fortunately, the mathematics permits a separation of the problem into a kinetic energy of translation plus a kinetic energy of rotation. I assume you know how to calculate these. All three objects will have the same kinetic energy at the bottom of the incline, but their rotational kinetic energies will all be different and their translational kinetic energies will all be different. The one with the greatest rotational energy is going to be the one with the greatest moment of inertia, so it will have the lowest translational kinetic energy and the lowest speed. You should be able to calculate the final speeds of all three objects if you know the height of the incline.
 
  • #6
the height is not given, only the things i mentioned are given. no numerical values of anything.
so if i find the relative translational kinetic energies, the one with the highest is the correct answer?
 
  • #7
Right. You are not given the height, but you could find an algebraic expression for final velocity in terms of the initial height. It is not needed for this problem. A relative comparison is all you need to do, and yes, the one with the highest translational kinetic energy is the one that gets to the bottom first.
 

Related to Rotational Inertia: Pipe Reaches Bottom First?

1. What is rotational inertia?

Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to changes in its rotational motion. It is calculated by multiplying the mass of the object by the square of its distance from the axis of rotation.

2. How does rotational inertia affect the motion of objects?

The greater the rotational inertia of an object, the more force is needed to change its rotational motion. This means that objects with higher rotational inertia will require more torque to accelerate or decelerate.

3. What is the significance of the "pipe reaching the bottom first" in rotational inertia?

This concept is often used as an example to demonstrate the effects of rotational inertia. When a pipe is released from a horizontal position, it will reach the bottom of its circular path first due to its higher rotational inertia. This is because the bottom of the pipe is further away from the axis of rotation, resulting in a greater rotational inertia.

4. How does the shape of an object affect its rotational inertia?

The shape of an object plays a significant role in determining its rotational inertia. Objects with a larger radius of mass distribution, such as a disk or hoop, have a higher rotational inertia compared to objects with a smaller radius, such as a rod or sphere.

5. What are some real-world applications of rotational inertia?

Rotational inertia has various practical applications in everyday life, including the design of vehicles, sports equipment, and amusement park rides. It is also important in understanding the stability and maneuverability of objects in motion, such as airplanes and satellites.

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