A question of physics for the mechanical enginner

It follows a hyperbolic curve, falling off faster at first, and losing less velocity per second as it slows.

Hyperbolic, getting shallower as speed approaches zero.

In the very common case where the force causing a decrease is directly proportional to the quantity still present, the curve is an exponential approach to zero. Rubber tires have a non-zero minimum rolling resistance due to deformation, so they stop abruptly from some low speed. So the real answer is to video it and measure the change per frame, to see which process dominates. On my grinder the solid-on-solid friction of the starting switch stops the wheel.

jsw

But, friction is not ONE constant. A fluid film bearing has several different forms of friction (mostly fluid shear) and rolling-element bearings have several different forms (mostly friction of the rollers against the spacer). The bearings may have sliding seals, there is air friction (windage) inside the motor and some magnetic drag, depending on motor type.

When the mechanical starting switch trips back on as the rotor slows, friction increases greatly (assuming a classic centrifugal starting switch). Those grinding wheels have a lot of inertia.

Jon

Depends... :)

As others have mentioned, it relies on the specific force balance in the system.

If it were a fixed, constant force the change in acceleration would be linear and the change in velocity quadratic.

If it were a purely proportional friction force alone it would be exponential.

In reality it's a combination of forces, some linear, some not. Which is dominant depends on the details of the system. For example, on the car, the wind resistance force is basically proportional to the speed squared while in a sliding body the frictional force is roughly linearly proportional to the weight of the body and the materials and is essentially constant (until static rather than dynamic friction effects dominate).

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If the force were constant, there'd be no change in acceleration (a=F/m), and the change in angular velocity would be linear.

A little bit of both.

Air resistance, which is not trivial, is exponential (not exactly, but you can think of it this way).

Bearing resistance is linear (constant resistance).

i

"It Depends" is correct, and, if the mechanical engineer in question is an independent consultant, "if I were under contract I could answer that for you (then bill for my time)" is both correct and pleasing (to him).

The two main forces on your grinder are probably going to be windage or other fluid drag, which acts as the square of velocity, and running friction, which is a constant.

So the speed will tend to drop off rapidly at first (I think it'll be close to a hypebola) while the windage dominates, then at the end it'll decrease at a constant rate while the running friction dominates. At the very end, particularly if you've got ball bearings that are a bit Brinelled or if things are a bit cockeyed, then the wheels will rock back and forth in a motion close to a damped sinusoid.

For the speed to start dropping slowly, then more rapidly as zero velocity is approached, you'd need an opposing force that increases as the speed goes down. With the exception of stiction (which only comes into effect when things are going really slow, and is pretty weird stuff to try to describe using physics) I don't know of any natural forces that do that.

f = ma and v = at

If the decelerating force is constant, the (negative) acceleration is constant and the dercrease in velocity is linear.

What are the decelerating forces? On the grinder wheel, friction and air turbulence inside the safety housing. Frictional force is presumably (approximately) constant. I can't estimate the magnitude of the air turbulence force: I'd guess that it was very small but not constant, i.e. greater at higher speed.

If there's a force that is a (non-constant) function of velocity, that force would change as the wheel slows. Are you saying, Ed, that the force from air turbulence is large enough, at RPMs of a grinder wheel, to have a significant efffect?

Hmmm.... Might there be inductive forces on the motor core due to some degree of permanent magnetism? Dunno. Doesn't seem like it would be a very big force if it's there at all.

It's not. In the case of the vehicle example, the air resistance decreases as velocity decreases. Considering windage on the pedastal grinder is a similar situation.

Other friction is more complex, but, generally, friction decreases with decreasing rotational speed. All effects combined, in most cases, display decreasing resistance with decreasing velocity or rotational speed. Thus, velocity is something that looks like a hyperbola.

Oops, I missed your other questions. MY grinder has a wire brush on one end, and yes, there is a lot of windage there.

Not much on an induction motor. A bit of it due to self-induction, if there's enough going on there to sustain some magnetism through eddy currents or whatever.

Windage and bearing friction are likely to be the main components. With the vehicle example, add rolling friction of tires. All exhibit exponential increases with increasing velocity, and vice-versa.

You can make this as complicated as you want, but the basic relationship of Velocity to delta-V, when coasting, is going to look hyperbola-like.

Yeah, I started out thinking about velocity and displacement and had a a much longer-winded posting going when I decided to chop it drastically and didn't edit it correctly... :(

Thanks for the catch.

With regard to either, the only way to know accurately is to make a measurement. It's going to be different for a car and a grinder though. For the grinder I believe that friction in the bearings will dominate, while for a car it will at high speed be wind resistance and at low speed rolling friction. Also depends on whether it's in neutral or with the clutch engaged and whether it's a manual or automatic transmission, and whether the brakes are or are not applied.

In other words it's not a simple question.

Definitely not a simple calculation, especially for the car scenario (not a good example for a simple comparison). A moving car has a lot more variables.. much more surface area, tire type and air pressure, differential/real wheel drive's ring gear driving the spiders and pinion and spinning the driveshaft in addition to the (manual or automatic) transmission components/bearings.

The motor and flywheel example (or motor without a flywheel since the motor's rotor has the same properties) would likely be fairly simple in comparison, at least. For the motor example it's basically bearing and air resistance (open frame motor or TEFC with or without an external fan) and surface finish on the flywheel (open flywheel or enclosed), and possibly, maybe a wee bit of residual magnetism in the rotor of an AC motor.

It depends on the nature of the friction acting on the object to slow it down. Aerodynamic drag (a major factor for an auto, still applicable for the grinder), rolling friction (auto tires) and others all have differing speed vs force (torque) characteristics.

Most sources of friction have a torque (or drag) vs speed characteristic that is either proportional or some power law of the speed. So the retarding torque will be higher at higher speeds and taper off as the auto/wheel slows down. The result is something that resembles an exponential decay. But the precise mathematical description is complex.