Calculating HP to spin wheel

[djg101]

I've been tasked with calculating the HP required to spin a turntable from rest to a desired RPM. I've tried calculating this out using two different methods, and have came upon two different solutions. I've went through my formulas multiple times but have not found why I am getting two different answers (one is double of the other).

Variables:
r = radius of turntable [m]
m = mass of turntable [kg]
ω = desired angular speed [RPM]; converted to rad/s using 2*pi/60.
t = desired time

Method 1: Energy

Erot = 1/2 * I * ω^2

This gives me my energy required in Joules which I then divide by the time to get my required power. Using the moment of inertia of a think disc (1/2*m*r^2), my final equation looks like this:

P = [(1/2)*(1/2)(mr^2)*ω^2]/t
= [(1/4)*(mr^2)*ω^2]/t

Method 2

For the second method, I used the equations below:

τ = I * α
P = τ * ω ==> P = I * α * ω

First I calculated my angular acceleration.

α = (ωf - ωi)/t = ωf/t

The equation simplifies to ωf/t since we are starting from rest.

I then input this into my power equation using the moment of inertia of a disc once again. This time I get:

P = (1/2)*(mr^2)*(ωf/t)*ωf
= [(1/2)*(mr^2)*ωf^2]/t

As you can see, my two results differ by the number in the denominator. I have the first equation being multiplied by (1/4) and the second multipled by (1/2). Does anyone know if I am missing any steps in one of these equations that might cause me to miss there being another 2 being multiplied. Any help would be appreciated!

 

[MintJulep]

I think method 1 gives you the average power while method 2 gives you the peak.

Both neglect bearing friction, which may not be negligible.

 

[BrianPetersen]

What you have found is exactly as one would expect.

If you are accelerating from rest by constant torque, the energy that you are delivering starts at zero and increases to a maximum. The average will be half the maximum.

If you are applying constant power, you will find that at the moment of starting off from a stop, your acceleration will approach infinity, which is not a realistic real world scenario.

The torque-based solution more likely coincides with a real world mechanism.

As mentioned above, this does not consider friction.

 

[TheTick]

Also complicating things is the motor torque vs RPM. Available torque from a motor changes with RPM. AC induction motors typically start with max torque at zero and get weaker as they go faster.