# Torque to roll up mats

SevenToFive
I volunteer with an athletics group that has some large mats that they roll out for practice and I started thinking of a way for us old timers to roll them up at the end of practice.
The mats are rolled up on a reel, that when empty is 14" in diameter, when full with the mats the diameter is 60 inches. I was thinking of adding a gearbox to the roll the reel. The mats are around 300lbs and around 150 feet long. We would only have to use the gearboxes to roll the mats back onto the reel, as it is really easy to unroll the mats by hand so I would have to find a clutch to engage to allow for freewheel option. The mats rest on a hardwood floor so friction should be pretty low.
To find the maximum torque needed would be 300lbs x 30in=9000in-lbs correct?

Thanks for checking my theory and answer and feel free to correct me if I am way off base.

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If friction was not involved you would not need any torque at all.

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Some minimum (not maximum) torque would be necessary even in the absence of friction, IF we want the rolling to be done with a minimum angular acceleration, and the maximum of (the minimum torques ) would be when the mats are fully wound on the reel cause that's when the moment of inertia becomes maximum.

BUT I guess the OP doesn't ask for that, he considers that the torque of weight has to be countered but it is actually the torque of friction that has to be countered in order to rotate the reel. The torque of weight would be zero IF the mats are made from constant density material so that the centre of mass is always the centre of rotation as the mats wound around the reel. if the mats are not of constant density then the centre of mass will not necessarily be the centre rotation hence the weight could have torque , and the torque of weight will be switching from positive to negative and so on as we rotate the reel.

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The mats being rolled up onto the reel have a stress induced on them. The torque necessary to make the curl in the mats could probably greater than the friction from sliding on the floor.

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Some minimum (not maximum) torque would be necessary even in the absence of friction, IF we want the rolling to be done with a minimum angular acceleration, and the maximum of (the minimum torques ) would be when the mats are fully wound on the reel cause that's when the moment of inertia becomes maximum.

But you just need to accelerate at some given point. The OP did not mention any time constraints. Of course you can solve it if you are given a time constraint, but I do not think that is what the OP was considering.

The mats being rolled up onto the reel have a stress induced on them. The torque necessary to make the curl in the mats could probably greater than the friction from sliding on the floor.
This might be true, but dependent on the construction of the mat (in particular its bending stiffness).

If friction was not involved you would not need any torque at all.
The mat starts out at lowest possible potential energy on the floor. It ends up wound on an axis that has a height h above the floor. The change in potential energy says there must be a torque to lift the mat.

SevenToFive
If friction was not involved you would not need any torque at all.
There has to be some sort of torque to turn the reel that will pull the mats up. Even if we roll up the 150ft in 5 minutes there has to be a torque generated...right? But how do I calculate that?

But how do I calculate that?
Like this ...
Assume the spool axis is 3 feet above the floor.
Mat weight is 300 lbs. Length is 150 ft. That is 2 lbs per foot.
There will be 3 ft * 2 lbs/ft = 6 lbs of unbalanced mat rising to the spool axis.
Maximum torque due to lifting the mat will be at the end of retraction when radius is 30”.
The torque will then be 30” * 6 lbs = 180 inch.lbs = 15 ft.lbs

The minimum lifting torque will be at start of retraction, 7” * 6 lbs = 42 inch.pounds = 3.5 ft.lbs

Maximum friction will be at the start of retraction when all the mat is being pulled across the floor.
Assume the "mat to floor" friction coefficient is 0.2, then 300 lbs * 0.2 = 60 lbs.
Radius of spool begins at 7” so torque is 7” * 60 lbs = 420 inch.lbs = 35 ft.lbs

The maximum torques operate at different ends of the retraction process.
So the maximum torque is really 35 ft.lbs + 3.5 ft.lbs = 38.5 ft.lbs.

Air is trapped below the mat during deployment so mat friction is very low while “floating” the mat down off the spool. After some use, before retraction, the mat will have lost that air layer so friction will be critical. To get a better estimate of the friction coefficient during retraction, use a spring balance to measure the force or torque needed to drag the mat after use.

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The mat starts out at lowest possible potential energy on the floor.
This is an assumption that was not stated in the OP. In general, this problem is probably very far from idealised in any way regardless i doubt that the torque due to lifting the mat is going to be the main contribution in the real situation.

There will be 3 ft * 2 lbs/ft = 6 lbs of unbalanced mat rising to the spool axis.
The underlying assumption here that the mat is hanging straight down is very likely not realized in the actual situation. The easiest way of doing this without making a lot of assumptions that we do not know whether or not they are reasonable would be to actually measure the required torque in the real situation.

The underlying assumption here that the mat is hanging straight down is very likely not realized in the actual situation.
I made the reasonable assumption that only the vertical component of the mat trajectory was due to gravity. I attribute any horizontal component as being due to friction, which is being ignored by other analysts.

malemdk
When you roll the mat on the reel, the mass moment of inertia increase , so the torque required also increases
its really a variable mass moment of inertial problem although your estimation is correct its not exactly right!

although your estimation is correct its not exactly right!
I agree.
I neglected the torque needed to overcome the initial static friction and accelerate the entire mat mass on it's slide across the floor. I also ignored possible strategic variations in the torque and the rate of rotation of the spool as radius increased.

There was insufficient data to give an exact answer. Any guesstimate needed to be an engineering approximation based on general assumptions. The question was posted in the “Mechanical Engineering” forum.

Post #7 demonstrated the frustration of the questioner with pedantics and demonstrated a need for a quick approximate answer. Post #7 also represented the last communication from the questioner.

Suffice it to say that this problem would never have a complete data set, so it could never have a rigorous answer as might be possible in a fully specified physics examination question.

OldYat47
Get a couple of heavy duty fishing scales. Attach them to the mat and have some help pulling on the scales until the mat started to slide. Have everyone observe the maximum reading on their scales. The sum of all the readings would indicate how much force it takes to overcome static friction. You could then use that number to calculate a conservative coefficient per unit area of the mat. Add 25% (just a number I pulled out of the air, should be reasonable) to account for unknowns. As the winding occurs the torque required will increase. You could use the friction per unit area (decreases as less mat is on the floor) vs. diameter of roller and rolled up mat to see what the maximum torque would be. Nowhere near scientifically rigorous, but it would probably work just fine.