Fundamental understanding of motor torque

I have the following elementary question regarding motors:

Motors provide mechanical torque as output. Every motor has a particular torque-speed characteristic curve which tells us how much torque the motor provides at a given speed at steady state once it has fully accelerated up to that speed.

Now lets imagine we rigidly attach some rotational inertia to the motor, such that the load and motor shaft rotate as one, and after turning on the motor and waiting for some time, the motor is rotating at some constant speed, say 1500 rpm. But according to the torque-speed characteristic curve, the motor will provide some torque corresponding to this speed, lets say 1 Nm. According to my understanding of physics, torque causes angular acceleration, which tends to increase the rotational/angular speed. So if the load is being supplied with some torque by the motor, then how is it rotating at a constant speed? Shouldn't it accelerate due to the motor torque?

I apologize if this question is too dumb. I am currently unable to wrap my head around this.

Real motors generally have a torque (measured at the shaft) which drops to zero at some speed for a given input. That is due to friction and "windage" (air resistance on the rotor) as well as motor characteristics. All the input power is being used just to keep the moving parts spinning against those losses (lost as heat in the bearings, in the motor coils and in moving air around).

In the case of a permanent magnet DC motor the back-EMF increases with increasing RPM so that the current and thus torque drops as the motor spins up. With no load and no friction it would increase to the RPM at which the back-EMF exactly equals the applied voltage. In reality, with no load, it's a bit less than that. So self-limiting RPM. With an inertial (not friction) load the general behavior would be exactly the same, just a different time scale.

Some real motors (eg. series-wound DC motors) may actually fly apart before they reach the RPM at which losses= input power.

At the risk of duplicating what others have answered, let me try.

According to my understanding of physics, torque causes angular acceleration, which tends to increase the rotational/angular speed.

Yes it does, but let's recast this in terms of something else.

Force causes linear acceleration, which tends to increase the linear speed.

Does that sound familiar? So, what happens to a car when you step on the gas? The engine produces an increased force which causes the car to accelerate. Why doesn't the car continue to accelerate until it flies off the earth? Because increased velocity produces increased friction/drag which opposes the force provided by the motor. At the point where the two forces cancel out, the velocity is constant.

In the case of your motor/load, torque causes angular velocity to increase. The increased velocity produces an opposing torque (drag and bearing friction). At the point where the torques cancel out, the angular velocity ceases to change.

You may be thinking, "But the air doesn't spin, so how can it produce a torque?" Well, any friction tangential to the axis of rotation WILL produce a torque. Good old "force times radius".

Yes, if the motor is applying torque and there is no energy lost (to mechanical loads, electrical losses, or just to friction), the shaft will accelerate. But there's always electrical losses and friction, and a motor running without some sort of mechanical load isn't very useful, is it?

Every motor has a particular torque-speed characteristic curve which tells us how much torque does the motor provide at a given speed

Yes and no: the torque speed curve is a rated limit, usually for continuous duty. In your example when the motor reaches equilibrium then the torque output is equal to the losses in the system (as others have detailed in other answers). Even in a lossless system the torque speed curve will hit zero (ie at synchronous speed for an induction motor) and that’s where your example would end up.

Edit: note that my comment on the curve being a maximum rating is more relevant to controlled motors, a line connected induction motor will follow that speed torque curve religiously, though my analysis on equilibrium remains the same.

So if the load is being supplied with some torque by the motor, then how is it rotating at a constant speed? Shouldn't it accelerate due to the motor torque?

the electromotive force produces a torque and there is an opposing torque due to friction of the rotor on the stator bearings. In turn, friction in the load, in turn friction of the mechanical part that is moving the motor. In short, the net torque is zero and therefore there is no change in angular speed (the motor arrives at a constant angular speed).