# smaller radius requires less torque?

I have been trying to build an electric go kart, and power requirements (as always) have been the problem. I have been going through some formulas to find out how big of a motor I would need from how much torque is required.

Formulas used:

• $F=ma$
• $M=Fr$

My specs:

• Weight of vehicle (driver included): 200kg
• Top speed: 20m/s (45mph) acc. (a)= 2m/s^2(10 sec to reach top speed) radius(r)=0.2 (variable tho, this is the problem)

From calculations, F=200kg*2m/s^2=400N, torque=400N*0.2m=80 Nm. If a smaller radius is used(say 0.1 m instead of 0.2 m), less torque is needed(400N*0.1=40Nm). Clearly, there is some misunderstanding, or we would all use smaller wheels if they resulted in less torque needed. What am I doing wrong? maybe I did not understand the formulas correctly? If I am correct, how would you explain using bigger wheel sizes?

• Change the title to “smaller radius applies less torque”, then re-think what you want to achieve... – Solar Mike Aug 28 '18 at 14:24
• Smaller radius has less inertia so its easier to turn the wheels. But wether thats good or bad depends on what you are optimizing. – joojaa Aug 28 '18 at 14:30
• Welcome on the Engineering SE! The editor box has many nice formatting elements, I suggest learn to use them. I improved the first half of your post, see how wonderful become it. :-) We have also Latex support for formulas (for example, type in $\sqrt{\frac{2s}{a}}$ and you get $\sqrt{\frac{2s}{a}}$), it is a little bit more complex but still easy. Well, and the universally accepted symbol of the torque is: M. Good luck on the site! – peterh Aug 28 '18 at 14:30

Welcome, Hey. a quick and easy way to get an estimate is like this:

Have you ever watched kart races in which the vehicles are powered by gasoline engines? Can you find out the horsepower rating for a gas kart whose speed and acceleration rate you wish to match? This is kind of a cheat (no equations required, woo hoo!) but it will get you started.

The next step is to decide on what you want the top speed of the kart to be; this then sets the overall gearing ratio for the drive system.

The desired range tells you how heavy a battery pack you will need.

In my own experience and as a calibration point, a 1.5HP gas kart is fairly doggy, a 3HP kart is almost fun, a 5HP gas kart has more zing and can carry two people, and a 10HP gas kart (especially a shifter kart) can go like stink.

At 746 watts to the HP (sorry about the equation), you can do the math!

• Good answer - kind of a cheat perhaps, but often this kind of route to a solution is more reliable! – Jonathan R Swift Aug 28 '18 at 19:25

The important requirement for accelerating the kart quickly is the power (work done per second) that the motor can deliver. The relevant equations are $$\text{power}= \text{force} \times \text{velocity}$$ for motion in a straight line, or $$\text{power}= \text{torque} \times \text{angular velocity}$$ for a rotating shaft.

If you use smaller wheels, the angular velocity increases for the same linear speed of the kart, and the torque is therefore smaller for the same power, but the motor has to turn faster.

If you changed the gear ratio to keep the motor speed the same, then the motor torque would be the same, though (because of the different gearing) the torque at the wheels would be less for the smaller wheels.