I took apart a sharper image personal fan run by a 9volt battery.
I understand a resistor is sometimes necessary, such as when preventing light emitting diodes from burning out. How about a 9volt motor run on a 9volt battery? Is a resistor necessary? Can a capacitor substitute for a resistor in some cases? (that was my guess as to why it has a capacitor) Is the function of this capacitor to slow the fan down, and if so does that mean running it straight to the battery might burn it out?
The function of the capacitor is to suppress electromagnetic noise (EMI) that the motor's coils will generate while in operation, due to the contact brushes inside the motor making and breaking the circuit (i.e. commutation). If you search this site, there are several questions covering motor EMI, so I won't detail it here.
Such EMI noise suppression is best done as close to the source of noise as possible. That is why you see the capacitor across the motor leads, and not further away. Even if the entire circuit consists of just the battery and the motor, EMI will radiate from this circuit, and other nearby electronics will be affected.
For instance, if the motor is operated very close to computer speakers with built-in amplifiers, you might be able to hear the noise caused by this motor EMI being radiated.
Regarding resistors: While resistors are often used for current control with LEDs, motors are best not controlled using resistors: The resistor would generate a fair bit of heat, wasting the energy that could be used by the motor.
Resistors are used for current sensing though, in some motor circuits.
This is a bypass capacitor. High frequency electronic noise generated by the arcing at the motor's brushes is short-circuited through the capacitor, which reduces the amount of that noise that is able to leak out and affect other circuits which are powered by the same battery.
simulate this circuit – Schematic created using CircuitLab
(The symbol for a motor is wrong: but CircuitLab doesn't have one!)
The capacitor forms an LRC low-pass filter together with the inductance and resistance of the circuit wiring and those internal to the motor. The current corresponding to the noise frequencies passes easily through C but not through the inductance and resistance, and so most of the noise voltage is dropped in the inductance and resistance and does not appear across C. If the noise does not appear across C, then it doesn't appear across the battery or the sensitive device. In a nutshell.
