This back EMF course says at 2:46 that in a DC motor, the operational speed is reached when the opposing magnetic force (induced by the back EMF) is equal to the motor effect (induced by the current through the armature.)
I don't understand why because if the two forces are equal and opposite and apply to the coil, it should stay stationary to me.
Think back to F = m•a but in a rotary situation: force is replaced with net torque. When the torque applied to the motor (the "motor effect" in the example) is equal to the torque produced by the motor, the net torque will be zero so the angular acceleration will be zero and angular velocity will be constant.
An ideal DC motor reaches speed equilibrium when the back-emf (induction due to rotation) equals the applied voltage at the terminals. At that point no current can enter the motor and therefore, there can be no torque on the rotor that would drive it to a faster speed.
For an non-ideal motor, friction losses mean that there must always be a net torque to keep the motor spinning at a constant speed. In this case the back-emf (due to rotation) doesn't quite equal the applied voltage and, this causes a current to flow into the motor. That current produces the torque to keep the motor speed in equilibrium (thus balancing the friction losses). Speed is slightly less compared to the ideal motor.
If you apply a mechanical load it's the same story as friction losses and, the motor slows to a lower steady speed. Again, that speed produces a smaller back-emf that allows more current to enter the motor. That produces more torque and, once again, equilibrium is reached.
the opposing magnetic force (induced by the back EMF) is equal to the motor effect (induced by the current through the armature.)
I don't know if you have misheard what they said or, whether they are using (somewhat) arcane sentences, but surely what I've written above your quote is the simplest way of looking at it. They seem to be over-engineering their words and missing the simplicity of the situation.
if the two forces are equal and opposite and apply to the coil, it should stay stationary to me?
For the "two forces" to be equal, the motor has to be rotating to generate the back emf that produces the (so-called) opposing magnetic force.
if the two forces are equal and opposite … it should stay stationary
Equal and opposite forces guarantee zero acceleration not zero velocity.
Be careful with the term EMF. Its dimensional units are voltage V=J/C. The force part appears when dividing by the distance over which it is applied N/C=J/C-m.
On the mechanical side, when the torque of friction is less than the applied torque, there is acceleration. When the torque of friction equals the applied torque there is zero acceleration (constant velocity).
The motor effect produces torque in response to electrical current. The generator effect produces voltage (back-EMF, \$E_b\$) in response to angular velocity.
On the electrical side, the applied voltage\$E_A\$ sets the maximum angular velocity. The internal resistance will drop some of this voltage by the amount of current flowing through it thus reducing “back-EMF” (angular velocity) attainable. The current (torque) required is determined by the torque of friction at that speed.
the operational speed is reached when the opposing magnetic force (induced by the back EMF) is equal to the motor effect (induced by the current through the armature.)
I disagree with that statement. The “magnetic force[torque,\$\tau_a\$]” is created by the motor effect \$\tau_a = k_{\tau}i_a \$. The back-EMF is induced by the generator effect from the angular velocity.
The armature current, \$i_a\$, thus torque, is actually determined by the lack of back-EMF. $$ \tau_a=k_\tau \frac{E_A-E_b}{R_a} $$
\$R_a\$ is the armature resistance.
