# Instead of a dynamo-meter, why can't we use $\tau=I\alpha$ to determine the power of a rotating shaft?

To measuring the power of a rotating shaft, dynamo-meters are used. Instead, why can't we find torque using Torque=Moment of Inertia times Angular Acceleration, $\tau=I\alpha$? And given that we know the rpm, $N$,

$$P=\frac{2\pi N\tau}{60}$$

Isn't this a more linear approach for power measurement?

You can. The problem is practicality, not physics. When you use a dynamometer, you can calibrate the meter once and use it for all the measurements you want. When you use the MOI you have to measure/calculate it for each measurement, as it is part of the unit under test. Sometimes you don't have access to the necessary data. Also measuring under acceleration means you don't have a steady state condition. You may want time to average the force at a given RPM so you get a better reading. You also don't want to exceed the allowable speed, which you may if you keep accelerating.

• But the MOI of shaft is going remain same always. right? And if the shaft is run by a lets say ac motor, its rpm will be same as long as voltage applied isn't changed. And the motor will attain a constant speed after sometime ie. a=0. so from v=0(rest) to instant before a=0(constant speed), acceleration can be measured. – Kameswaran Ganesh Feb 23 '15 at 18:22
• The MOI of the shaft will remain the same, but there might be coils attached that are spinning as well. Computing their MOI may be hard. Whether or not the motor attains steady state before something falls apart depends on the motor and setup. Yes, you can measure acceleration during that period. I didn't say it cannot be done, I suggested some practical difficulties that I think will mean the data is not as accurate as with a dynamometer. In some cases yours will be a good approach. – Ross Millikan Feb 23 '15 at 18:45

There are cases where you can do as you suggested and where you can't:

• A: If the external forces exerted on the rotating shaft doesn't change as the shaft accelerates, then you can simply apply the formula you suggested: Measure the MoI once, and use it at all times. Since the external forces are constant (or zero), your MoI will not change. Example: You have raised the wheels of a car so they they don't touch the ground and you hit the gas pedal: Wheels get accelerated freely. Your formula is applicable. Another example is flywheel energy storage devices when you are charging them. During charging no external electrical load (and a counter EMF) is present and your formula is again applicable. In all these examples, MoI of the car or flywheel wheels have constant effective inertia as experienced by the rotating shafts.

• B: There are external forces that are not constant i.e., during the measurement period they vary in a hard-to-measure way. Effect of non-constant external forces on the shaft are experienced as varying MoI. This will prohibit usage of the formula you suggested. Example: A regular car accelerating as it travels (wheels are touching the ground). Inclination of the road and the air resistance are all external forces which will be exerted on the shaft eventually. They all cause the effective MoI of the wheels to vary and your formula becomes useless. For the flywheel device (or an electricity generator in the very similar fashion), if there are connected electricity consuming devices, shaft will "feel" these loads in the form of a differing MoI.