I am thinking about buying a 12V DC motor but want to keep everything, including the motor, battery, and power switch somewhat compact. I am not very knowledgeable about these kind of things and am curious whether or not my 12V DC motor would run at all on a 9V battery.
According to your link the motor specs are:
- Rated Voltage: DC 12V
- Reduction Ratio: 1: 172
- No-Load Speed: 30RPM
- Rated Torque: 14Kg.cm
- Rated Current: 1.1Amp
With a permanent magnet brushed DC motor, speed is proportional to voltage and torque is proportional to current. Your motor is specified to produce 30rpm at 12V with no load, and 14kg.cm (presumably) at 1.1A. No-load current draw and rpm at rated torque are not specified.
A typical PP9 Alkaline battery produces ~7V at 1A when fresh, dropping to 5~6V after a few minutes. At 1.1A you might only get 4 minutes run time before the battery goes flat, at about half normal speed due to the lower voltage.
If the torque load is less the motor will draw less current, so you should get longer run time with a light load. However without a no-load current spec we have no way of knowing what the maximum run time could be. We also don't know how accurate the specs are. The only way to find out what it can really do may be to test it.
If you reduce the voltage to 9 volts, the speed will be reduced proportionally. The torque capability will not be reduced. If the battery is too small, it will be discharged quickly and the internal resistance may reduce the voltage significantly below the nominal battery voltage.
The rated motor current is 1.1 amps and the rated torque is 14 kg-cm. That means that the motor will require 1.1 amps to drive a load that require 14 kg-cm to turn. The motor should be capable of operating at the rated speed and torque continuously without failure for the normal lifetime. Unfortunately, the rated speed of the motor is not given. It is likely considerably less than the 30 RPM no-load speed.
A small motor with a gear reducer that provides a lot of speed reduction is not going to be very efficient. A lot of power will be lost in the winding resistance and in the gear friction.
A search of battery specifications should allow estimation of motor operating time and the minimum battery size. Battery life is given in ampere hours or milliamp hours.
See also: Calculate battery life
Not for long! (And the speed would decrease as the battery wears out and the voltage falls)
9V batteries are pretty "weak" because the other half of the story is how much current they can supply at that voltage (Power = Voltage * Current). If you want to spin a big motor, the battery voltage will sag because of the battery's internal resistance. The lower the supply voltage, the slower your motor will spin. Ultimately the battery and motor will reach an equilibrium speed for the load you have on it. Check out this page to see how the 9V batteries hold up under load (hint not well).
For reference, Adafruit's 130 size hobby motors consume only 70mA with no load (battery will last ~5 hours but won't be very useful) and up to 500mA when stalled (worst case scenario and if the motor doesn't burn up, the battery will last ~20 minutes).
As you can see, the alkaline battery chemistry is not designed for rapid discharge. A NiMH, PbA or LiPo battery would be a better bet if you need more power (and you could recharge the battery for more fun in the future!)
In principle, it's probably possible to run a 12V motor on 9V -- it'll just run slower and with much less torque.
But if by "a 9V battery" you mean a typical PP3 size battery (i.e, the rectangular ones with a snap connector on top), the answer is almost certainly "no". These batteries are designed for low-current applications like handheld radios, remote controls, or smoke detectors; they cannot supply much current, and will be unable to drive a motor for long, especially a motor that's designed for a higher voltage anyway.