An inclined conveyor belt, gravity, and rolling shapes of differing types

[tadietz]

I have looked around several internet Physics sites and information sources, and even asked questions here and in other places, and can't seem to find answers to my questions about a thought experiment I came up with:

Think about three similar cases: A ball, a solid cylinder, and a hollow cylinder, all with identical mass, rolling individually down an inclined conveyor belt rotating in an 'uphill' direction. Assume there is a sufficient coefficient of friction (CoF) between the objects and the belt to have them roll without sliding and that the conveyor belt's motor speed can be adjusted. Some questions about the system described come to mind:

1) Will the motor driving the conveyor belt do less work in an inclined vs a horizontal position to drive each object to some specified (> 0) rotational speed if the objects are placed on it with initial rotational speed = 0?

My intuition says yes, because an object rolling down a fixed ramp acquires its rotation only from gravity acting on it, and in this scenario, that same force is present and acting on the objects, in addition to the rotation being imparted to them by the conveyor belt's movement.

So, if this is correct, are there differences in the amount of work required by the motor to get each of the differing types of objects (ball, solid cylinder, and hollow cylinder) to the same rotational speed? That is, due to the fact that each object has a different moment of inertia and would accelerate down a stationary ramp at different speeds, does this have an analogous effect in this scenario and cause differing amounts of work to be required of the motor in order to get the objects to a specified rotational speed?

2) Next, is it possible to drive the above described system (regardless of the answer to question #1) to an equilibrium state, i.e., a point where each object would appear stationary with respect to its position on the inclined rotating belt from the perspective of a stationary observer viewing the system?

Again, intuitively it would seem possible, since even an object falling straight down reaches a terminal velocity, i.e., where the drag of air acting on the falling object counters its acceleration due to gravity until a constant speed is reached - at least in a non-vacuum. In my scenario, the drag would mostly be due to the CoF between the objects and the belt, while the acceleration to be offset would be the reduced rate of acceleration due to gravity
acting on the objects on an incline (standard ball-on-ramp calculations apply here, I am guessing, to calculate this rate of acceleration).

As with question #1, if my intuition is correct and an equilibrium can be reached, I am curious if there would be any advantage in terms of object shape to lessen the amount of work required to drive the system to equilibrium.

I hope my descriptions and analogies are sufficiently clear. What I would like to be able to come up with (assuming the answer to question 2 is 'Yes') is an equation that accurately describes the ball/inclined conveyor system which allows me to predict equilibrium states with differing conveyor inclines and speeds, ball sizes and masses, CoFs, etc., so that multiple combinations of factors that result in equilibrium states can be predicted.

Thanks for any thoughts on this.

 

[rcgldr]

1) Will the motor driving the conveyor belt do less work in an inclined vs a horizontal position to drive each object to some specified (> 0) rotational speed if the objects are placed on it with initial rotational speed = 0?

In the inclined case, the motor can do zero work, not move at all, and the object will roll down the slope, eventually achieving the targeted rotational speed. In the horiztonal case, the motor will have to do work.

2) Next, is it possible to drive the above described system (regardless of the answer to question #1) to an equilibrium state, i.e., a point where each object would appear stationary with respect to its position on the inclined rotating belt from the perspective of a stationary observer viewing the system? Would be any advantage in terms of object shape to lessen the amount of work required to drive the system to equilibrium.

To keep the object "stationary", the motor has to continously accelerate the conveyor belt so that the linear force on the rolling object = m g sin(θ). The lower the angular inertia of the rolling object, the faster the rate of acceleration would be in order to generate that force. In all rolling cases, the amount of power involved would increase linearly with time, since the force would be constant, with the speed increasing linearly with time. The amount of work done versus time is greater if the angular inertia of the rolling object is lesser.

For the inclined case, in order to keep the rolling object at a fixed point:

hollow cylinder: belt acceleration = 1.0 g sin(θ)
hollow sphere: belt acceleration = 1.5 g sin(θ)
solid uniform cylinder: belt acceleration = 2.0 g sin(θ)
solid uniform sphere: belt acceleration = 2.5 g sin(θ)

 

[tadietz]

In the inclined case, the motor can do zero work, not move at all, and the object will roll down the slope, eventually achieving the targeted rotational speed. In the horiztonal case, the motor will have to do work.

Thanks, rcgldr. I understand the stationary belt case, but meant to specifically exclude that condition by specifying in my description:

...rolling individually down an inclined conveyor belt rotating in an 'uphill' direction.

I should have been more explicit that the stationary case was excluded. What you state - that the motor can do no work at all if the conveyor is not rotating (i.e., is acting just like a ramp) - was sort of what led me to my thought experiment in the first place. I wondered if gravity can in any way 'boost' the output of the motor in the case of an object rolling down an inclined and rotating conveyor belt.

To keep the object "stationary", the motor has to continously accelerate the conveyor belt so that the linear force on the rolling object = m g sin(θ). The lower the angular inertia of the rolling object, the faster the rate of acceleration would be in order to generate that force. In all rolling cases, the amount of power involved would increase linearly with time, since the force would be constant, with the speed increasing linearly with time. The amount of work done versus time is greater if the angular inertia of the rolling object is lesser.

For the inclined case, in order to keep the rolling object at a fixed point:

hollow cylinder: belt acceleration = 1.0 g sin(θ)
hollow sphere: belt acceleration = 1.5 g sin(θ)
solid uniform cylinder: belt acceleration = 2.0 g sin(θ)
solid uniform sphere: belt acceleration = 2.5 g sin(θ)

I agree that the belt normally would need to continuously accelerate since the objects would tend to do so as well - but only when rolling resistance/CoF of the objects on the moving belt is not enough to counter acceleration due to gravity. That was sort of the thought experiment goal behind the second question, i.e., deriving a model that allows you to vary angles, masses, coefficients of friction/rolling resistance values, etc., to come up with something like the equation used to calculate terminal velocity in fluid dynamics.

I knew such a model had to account for the differing acceleration values of rolling objects of various types to predict the equilibrium conditions I was after. Thanks for supplying the equations, by the way - I remember learning that a solid sphere would accelerate fastest and the hollow cylinder the slowest because of their differing moments of inertia, but couldn't remember the exact formulas; you saved me looking them up.

In looking at all of this, I think I may be able to sort of combine the fluid dynamics terminal velocity equations and the ramp and rolling object ones and set it up to allow solving for the CoF needed to make the net acceleration 0. Then, of course, all I'd need to do would be to find materials for the rolling objects and the belt that have the right properties to make it work according to my model, then build it to see if I got it right

 

[rcgldr]

Thanks, rcgldr. I understand the stationary belt case, but meant to specifically exclude that condition by specifying in my description

Still you could use any constant and very low non-zero speed and get a similar low work result. Unless the motor moves the belt "downwards", gravity is always assisting in increasing the rate of rotation of a rolling object on an inclined belt.

rolling resistance

Including rolling resistance won't help much, since it's independent of speed, at least in the idealized case. It just decreases required force to something like .99 m g and decreases the required accelerations shown above by .01 g. If the rolling resitance coefficient = sin(θ), for example, rolling resistance = .01 = sin(θ), then any constant speed of the belt will keep the object "stationary".

 

[tadietz]

rcgldr,

I made a mistake using the term rolling resistance; I guess, doing a little review, the most appropriate term would have been coefficient of static friction since the belt and the objects rolling on it aren't slipping. Not that this matter now - I had an epiphany! That's the beauty of discussing ideas on forums with more knowledgeable folks, I guess, and then going off all over the Internet to try to educate yourself about things I either never knew, or forgot in the 30 years since I had physics in college.

My mental image of what I wanted to get to and the forces involved was clearly off. I was getting ready to write a long and involved post, relating what I thought would happen to the terminal velocity formula from fluid dynamics. To help with that, as I was reading about friction and just visualizing what would happen, I started equating drag coefficient to coefficient of friction in their effect. As I got into constructing my post, I realized that traditional frictional forces between the the belt and the rolling objects would never balance each other like the drag coefficient in the terminal velocity formula does to the acceleration due to gravity of a falling object; rather, the belt moving would just impart additional rotational energy to it, of course. Re-reading your initial response helped, too.

Anyway, from Wikipedia:

where

Vt = terminal velocity
m = mass of the falling object,
g = acceleration due to gravity,
Cd = drag coefficient,
ρ = density of the fluid through which the object is falling, and
A = projected area of the object.

What I need is a covering on the belt, object, or both that is just adhesive enough to produce an attraction between the belt and objects to counter the objects tendency to accelerate downhill.

So, if I am right in my thinking, I can adapt the fluid dynamics formulas so some factor relating to adhesion (coefficient of adhesion? hmmm) replaces coefficient of drag, and the rest of the factors in the terminal velocity formula can be used as is or ignored except for g, of course; g needs to be adjusted per your formulas for the angle of the ramp and the type of object so that the acceleration due to gravity is appropriately substituted into the terminal velocity formula.

For my purposes, I think I can just drop ρ since we are talking about air, but I'm not quite sure regarding A. I would lean toward dropping A as well, since neither a contact point (in the case of any spherical object touching the belt) or a contact line (for a cylindrical one touching it) have any area. Contact area would seem - more or less - to correspond to projected area in my adaptation of the terminal velocity formula now that I am thinking right about needing an adhesive force.

My only reservation against dropping A is that there needs to be a contact area for adhesion to occur, and if there were any deformation of the contact surfaces involved -and there might be some, especially with softer, stickier surfaces and larger masses - there could be some reasonably-sized contact areas produced.

I don't have a clue how to calculate their sizes, though. Intuitively, a hard sphere on a softer belt would make some sort of shallow, round dimple; a hard cylinder would tend to produce a shallow, rectangular, rounded-bottom trench in a softer belt. Other combinations of contact shapes can be imagined based on the relative hardnesses of the materials coating the rolling objects and/or belt - but it makes my head hurt to think of the math needed to figure out the area of these shapes, and I'm not even sure that contact area is analogous to projected area anyway.

What do you think?

 

[rcgldr]

I think that the adhesive factor would also end up being independent of speed.

 

[tadietz]

Sorry for the long delay - life and work stuff intruded on my having time to think about this for a while - and thanks for the earlier responses.

Anyway, I agree that in my system as initially described, the belt would have to continuously accelerate to keep up with the rolling object due to gravity's accelerating effect. That is, unless some force acted on the rolling object to counteract said accelerating effect, much like wind resistance limits a falling body to a terminal velocity.

So, thinking analogously, could you could perhaps apply some similar counteracting force on the rolling object to keep it from accelerating beyond a certain speed - say by pointing a compressed air stream at it once it has reached a certain speed (blowing uphill, of course), or having some sort of fins on the edge of the belt that either generate a moving air flow or mildly impede the motion of the rolling object through slight contact?

I would appreciate your thoughts.

 

[tadietz]

One more thing - does the rolling object in my system impart any 'uphill' force at the point of contact to the conveyor it is rolling downhill on? If so, how do you quantify that force?

Intuitively, it would seem that it would, kind of like a tire on the drive wheel of a car can propel a rug placed in front of it underneath and out behind it on a really smooth garage floor - especially if it has a 'grippy' backing, slick fibers, and is upside-down.

In my system, the rolling object is like the tire, and the conveyor is like the rug. I know the conveyor belt is moving, but that shouldn't change the answer, should it?

 

[tadietz]

Long time, no activity - but I have something to share relating to some pretty nice software I found, and I have some new questions:

I am playing around with some physics simulation / education software called Algodoo (http://www.algodoo.com- not too sophisticated for a casual user but has some scripting capability if you want to get more sophisticated and it is really inexpensive - like $3.99 for an individual license. It has some pretty nice capabilities for building up reasonably simple '2D' mechanisms and even to some extent '3D' in that you have multiple collision layers that you can specify objects interact with, as well as the ability to move objects in front of / behind other ones by various means. It is a bit 'colorful' in terms of the backgrounds, colors used, etc., given that it is geared for the pre-university level education market, but the physics engine behind it is supposedly the same one used in a much higher-level package from the parent company, Algoryx.

So, I decided to model the original situation I opened this thread with, and was able (using two approaches outlined below) to keep various sorts of circular objects stationary by accelerating the conveyor belt moving uphill to keep up with the various objects' acceleration due to gravity down the belt due to its slope.

You can set and find out lots of information about various of the objects' properties before and during the simulation, but one thing I was curious about I can't seem to get a bead on is the total energy the motor has applied to the system up to whatever point I decide to stop it. All I have to go on with respect to the motor's parameters is the motor RPM and the torque, both of which are dynamically adjustable during the simulation.

Adjusting either torque or RPM during the runs can get you to the desired state, e.g., setting the RPM very high and the torque low but increasing torque during the run, or setting the torque high and increasing the RPM slowly during the run. Either way allows me to keep the object rolling downhill stationary with respect to the observer for quite a while.

Anyway, given all the above, is there a way - knowing all the masses of the objects, speeds of rotation of the rolling objects and belt, time spent in the run, RPM and torque of the motor, etc. - that I can calculate the energy used by the motor up to the point I pause/stop things? I have poked around here and other places but didn't find anything obvious - although I could have used the wrong search arguments.

 

[rcgldr]

In watts, power = torque (N m) x rpm x 2 π / 60.

Let p(t) = the amount of power generated at time t. The then the total energy produced by the motor will equal the intergral of p(t) dt from 0 to some elapsed time t₀.