I do not have a ground connection to the Arduino, is it necessary? I am new to electronics, could you tell me why?
Yes it is necessary, because the gate voltage for the mosfet is coming from the arduino and it won't be a circuit without a ground return for the gate drive signal back to the arduino. You can't light a lamp by just having one wire from the battery positive to the lamp without also having a wire from the other side of the lamp back to battery ground. The mosfet source terminal is the return path back to whatever is driving the gate terminal.
The MOSFET is IRF520, I know it wont be fully activated by the Arduino since it is not a logic level MOSFET but it is enough for my purposes since me motor draws only 70mA
Lastly thanks for the help!