# Homework Help: Power Expended by a Fan Cart

1. Dec 8, 2015

### IMAGINETHINGS

1. The problem statement, all variables and given/known data

I stumbled upon some thought, suppose we have an electronic fan cart like this https://i.ytimg.com/vi/Zy7JQ0lwj1w/maxresdefault.jpg

My question is, when I turn it on and it continue to move, is the power expanded by the fan cart constant?

I suspect it is not because it is accelerating, but based on my researches in the internet, the majority of the reports I've read said that it is constant?

2. Relevant equations

3. The attempt at a solution

Based on my researches, almost all answers about this question said that the POWER is CONSTANT because of the BATTERY, which did not convinced me because I know that a battery supplies a constant voltage (not power W) and the power (p=vi) is also dependent on the current passing through the motor (and according to my research as the load of an electric motor increase, the current passing through it also increase, therefore power increases).

I also tried to get the derivative of the Power equation P = F*v, to see if the Power changes, which lead to dP/dt=F*a (which is non-zero, therefore the power changes)

Also from the equation itself that, P=F*v, we will also see that velocity increases so Power also increases.

Am I right? Is power not constant in this thing?

2. Dec 8, 2015

### Buzz Bloom

Hi IMAGINETHINGS:

I am not an expert on motors. As a general understanding, I believe that you are right in that as load increases resistance/inductance increases, so power increases. However, since most of the sources you found say that the power is constant, I would interpret this to mean that after the switch turns on the motor, it very quickly reaches a state where the load will generally become constant, and it will remain constant unless some kind of not normal operation (like maybe jamming the motor) takes the motor out of its steady state load operation.

During the constant load state, the nature of the load will change. At first, the boat will be accelerating at the maximum rate, but as water and air resistance increases (as I vaguely remember, proportionately with velocity squared), the acceleration will decline until it becomes zero. After this, all the power will be in maintaining a force to balance the resistance at the maximum velocity.

Hope this helps.

Regards,
Buzz

3. Dec 8, 2015

### IMAGINETHINGS

As I had understood in your reply, you're saying that (in the context of your boat example) the power of the motor of the boat is not constant while it is accelerating, but as the force from the boat's motor achieves dynamic equilibrium with the external forces (thus the boat moves in constant velocity) the power becomes constant. Did I get it right?

Regards,
IMAGINETHINGS

4. Dec 8, 2015

### Buzz Bloom

Hi IMAGINETHINGS:

I apologize for not being clear. My thought is that during the short period of time that the fan it getting to its operational speed, the load increases from zero to its maximum. Then the load remains constant during the period of acceleration, and then also while the velocity is constant at its maximum. I guess that there may also be a mechanism for the fan to adjust its speed and/or its blade pitch as the air flow through the fan increases with the velocity of the boat. I do not know whether such a mechanism is necessary to maintain constant load.

Regards,
Buzz

5. Dec 8, 2015

### Merlin3189

Now that someone has broken the ice, I'll chip in my random thoughts.

Constant power can obviously provide acceleration, but the force needs to decrease with velocity and eventually a constant speed equilibrium can be reached.

But what power are we talking about? The battery supplies power to the motor, some of which is dissipated as heat (&c) and some to the prop. The prop produce force on the air and on the cart. So some power ends up driving the cart and some power warms the air.

If the cart velocity is always small compared to the air velocity produced by the fan, then I think the thrust will be nearly constant and the power given to the cart will be proportional to the velocity. But the fan/motor speed may be nearly constant, so the power input to the motor will also be nearly constant.

As far as motors go in general (at least the permanent magnet DC motors as shown in the illustration) the torque is inversely proportional to the speed, so the power output (torque x speed) has a maximum near the middle of its speed range. So with a simple fixed voltage supply you only get constant power if you have constant speed.

Normally PMDC motors with fixed voltage supply are operated near their max efficiency rather than near their max power. This is near the top of its speed range, where the torque is low (because the current is also low) and power varies greatly with variations in speed. If you operate near the max power, the motor is rapidly going to overheat.
However, if you use a lower voltage, then it can be run nearer to max power (abeit a lower max power) where the variation in power with speed is minimum. So this cart may be using a 6V motor (one designed to run near max efficiency on a 6V supply) but running it on 1.5V near its max power point.

Net conclusion: it is possible that the power input to the motor could be made nearly constant by judicious choice of motor operating point. Just picking any small motor, putting on a handy little prop and connecting up a 1.5V battery is unlikely to land you at that point.