What if a ramp were covered in rubber?

Ice is very slippery. Things slide on it easily.

Rubber is grippy. Your rubber boots help you not slip. What if a ramp were covered in rubber?

Put a block at the top of a ramp and let go. Will it slide? That depends on two things: how steep the ramp is, and what the ramp and the block are made of.

In this lab the ramp is 4 meters long and tilted at a gentle 15°. First it is ice on ice, a very slippery pair. Then we cover it in rubber against dry concrete, the grippiest pair in the textbook's table. Will the block still slide, or will rubber hold it still?

Friction between two surfaces depends on the materials, summed up in two numbers: the coefficient of static friction μs, which limits how hard friction can hold an object still, and the coefficient of kinetic friction μk, which sets the drag once it slides.

This lab keeps the slope fixed at 15° and swaps the material pair in the friction-slide model, from ice on ice (μs 0.10, μk 0.03) to rubber on dry concrete (μs 1.0, μk 0.7), both from OpenStax's Table 5.1. Then it asks how steep each ramp can get before the block starts to move.

First the ramp is icy. Olly lets go of the block.

It slides down! Then it glides a very long way across the icy floor.

In the model, the icy ramp is tilted at 15° and the block is let go. It slides at once, speeding up by about 2.3 meters per second every second. It reaches the bottom moving at about 4.2 meters per second.

The floor is icy too, so very little friction slows it down. The block glides about 31 meters before it stops, more than seven times the length of the ramp. Ice is so slippery that the block would slide on almost any slope: in the model, it starts to slide on an icy ramp steeper than about 5.7°.

At 15° on ice the model gives a = g(sin θ − μk cos θ) = 9.81(0.259 − 0.03 × 0.966) = 2.26 m/s². Over L = 4 m the block reaches v_b = √(2aL) = 4.2 m/s, then decelerates on the icy floor at μk g = 0.29 m/s² and slides v_b²/(2μk g) = 30.6 m.

The holding limit is arctan 0.10 = 5.7°. Check it: at 5° the model reports "stays put"; at 6° the block slides with a = 0.73 m/s² and still glides 10.0 m across the floor.

Now the ramp is covered in rubber. Olly lets go.

The block stays still! The rubber grips it tight. Olly has to tilt the ramp very steeply before it slides.

Now cover the same 15° ramp in rubber. Let go, and nothing happens: the block stays put. The rubber's grip pushes back up the slope just as hard as gravity pulls down it.

How steep can you go? In the model, rubber holds the block still all the way up to 45°, while ice gives up at about 5.7°. At 46° the block finally slides, but the grippy rubber floor stops it after only about 1.3 meters. Even a hard push does not get far: a block pushed at 3 meters per second down the 15° rubber ramp slows down and stops after about 1.1 meters.

With μs = 1.0 the block holds while tan θ ≤ 1, i.e. up to 45.0°: the model reports "stays put" at 15° and at 45°. At 46° it slides with a = 9.81(0.719 − 0.7 × 0.695) = 2.29 m/s², almost the same as ice at 15°, reaching 4.3 m/s; but the floor decelerates it at μk g = 6.87 m/s², so it stops in 1.3 m rather than 30.6 m.

Given a 3.0 m/s push at 15°, the rubber block's acceleration is negative, a = 9.81(0.259 − 0.7 × 0.966) = −4.09 m/s², so it stops on the ramp after v₀²/(2|a|) = 1.1 m. The same push on ice sends it 45.9 m across the floor.

Gravity pulls the block down the ramp. Grip pulls back.

Rubber grips much more than ice. So the pull has to be very big before the block slides.

On a ramp, part of gravity's pull points down the slope. Friction can push back, but only up to a limit. That limit depends on the two surfaces. For ice on ice it is small, so even a gentle slope beats it. For rubber on concrete it is about ten times bigger.

OpenStax, a free physics textbook, explains that part of friction comes from the molecules of the two surfaces sticking to each other, which is why rubber-soled shoes slip less than leather ones. Our model uses one typical number for each material pair and ignores air, wetness and wear.

A block at rest holds as long as the needed friction, mg sin θ, stays below the maximum static friction μs mg cos θ; dividing gives tan θ ≤ μs, with mass canceling. OpenStax uses the same balance on an incline, written for constant-velocity sliding, to measure μk = tan θ.

OpenStax attributes part of friction to adhesion between surface molecules, which varies with the substances in contact, and notes that coefficients are given to only one or two digits because the simple friction law is approximate. The model inherits those limits: Coulomb friction independent of speed and contact area, one coefficient per pair, no tipping, the same surface on ramp and floor.

Ramps for wheelchairs are gentle. Their surface must not be slippery.

Rubber soles on shoes help you grip the ground too.

Builders think about grip all the time. The U.S. Access Board's guide to the ADA accessibility standards says ramps may rise at most 1 unit for every 12 units along, a slope of less than 5°. It also says floor and ground surfaces people use must be stable, firm and slip resistant.

Interestingly, the standards do not set a single grip number, because there is no widely accepted way to measure slip resistance. In our model, an icy ramp could barely hold a block at that slope, while a rubber one would hold it with room to spare.

The U.S. Access Board's ADA guide limits ramp running slope to 1:12 (8.33%) and recommends less where possible; accessible floor and ground surfaces must be stable, firm and slip resistant. It also explains that the standards do not specify a minimum coefficient of friction, because no widely accepted test method has emerged and ratings depend on the test protocol.

A 1:12 slope is about 4.8°, just under ice's holding limit of 5.7° in this model and far below rubber's 45°. Real slip resistance changes with water, dirt and wear, which is why the guide notes that slip-resistant finishes may need upkeep.