# Wheel size and speed help

Wheel size and speed help!

Okay, my father and I have been an argument for days on this. We were arguing about wheel sizes and their effected distance traveled. A larger wheel has a greater circumference; therefore one revolution will travel a farther distance. Here's the argument. If you have two wheels of different sizes, one small and one large, and rotate them with the exact same force on an axle- Will the larger wheel travel a farther distance in a given time because it can cover more distance, or because it actually rotates faster at the point of contact on the ground. My argument is the smaller wheel and the larger wheel can actually be "spinning" (rotational speed) at the exact same speed, but because a larger wheel will cover a greater distance on one complete turn, will move faster along a plain. I understand that if they spin the same speed, the smaller wheel will have made a complete turn before the larger wheel. It is my belief that even though this is true, the larger wheel has still made enough distance (even though it has not yet made a complete revolution) to eventually cover more ground; thus, creating more distance; thus having a faster linear speed. So I guess after all that mumbo jumbo, I'm trying to ask do two wheels of different sizes travel at different linear speeds with the same rotational speed at the axle because of size, or because of the speed of the two wheels (at the point of contact on the ground) actually changes (rule out all variables: surface friction, wind, etc.)? I hope someone can make sense of all of this and put a stop to our bickering :).

P.S. Also, If I took two wheels, again of different sizes- one large, one small, and started them on top of a slope. If I let go of both wheels at the same time (these wheels are the same weight) would the larger wheel reach the end before the smaller wheel? Or would they reach at the same time? Also, let's say that the slope was long enough for each wheel to reach maximum speed. Please explain the answer to this.

## Answers and Replies

E8
I'll take a shot. I believe that as you increase the diameter of the wheel you increase the inertia (Inertia = 1/2 mass * radius ^2). Because you stated that both wheels receive the same amount of force I believe that the smaller wheel will have a greater acceleration than the larger wheel since the smaller wheel will have the least amount of inertia ('laziness') and thus require the least amount of force to get it going. The larger wheel will travel the greater distance and that is the trade off when you increase or decrease size you trade distance for force. I know for sure that the large wheel does travel a longer distance per revolution than the smaller wheel but the 2 wheels are not traveling at an equal revolution/unit of time.

To your 2nd question I believe that the smaller wheel will beat the larger b/c you stated that the wheels were the same weight (mass when gravity is applied) and again I = mr^2, if the mass remains constant and the diameter is increased the inertia and amount of force needed to move the larger wheel increases.

I'd wait for somebody else to confirm or correct me before you do anything I'm just Microbiologist studying physics for the MCAT and reading these boards is one of my favorite ways of studying. I think the bigger question is why you're trying to show up your dad so close to father's day.

It's the greatest gift I could have gotten him. He loves to argue :).

Staff Emeritus
Gold Member
Honestly I'm not sure I completely understand what you're asking, even after having read the question several times. Perhaps you should spend a little more time formulating your questions so they are easily understandable.

What I understood is that you want to put two wheels on either end of an axle; one wheel is smaller than the other. You want to know what will happen. The answer is that given the same angular velocity, the larger wheel will cover more linear distance on the ground. You can build an example out of Legos quite easily. You could also go let some air out of one of the front tires on your car (I don't recommend this, of course, it's dangerous) and discover that your car pulls in the direction in the deflated tire.

I should also mention that the point of contact of a wheel on the ground is NOT MOVING. The wheel rim and ground have zero relative velocity in normal rolling motion. The only time the point of contact is not stationary is when the wheel is sliding.

The second question was answered properly by E8. The wheel with the smaller rotational inertia will reach the bottom first.

- Warren

Homework Helper
I understand that if they spin the same speed, the smaller wheel will have made a complete turn before the larger wheel.

There's your problem. Since you specifically said that by "spin at the same speed" you meant the same angular velocity, both wheels will make a complete turn (2 [pi] radians) in exactly the same time.
The larger wheel, with greater circumference, will cover the greater distance. If you have two wheels of different diameter attached by a rigid axle, they will move in a circle with the larger wheel on the outside of the circle.

Zantra
I don't see what's so hard to understand. He's asking that if you have say, 2 carts side by side, one having a larger set of wheels than the other, and they each have the same forward force applied, which one will travel farther. Now we have to also clarify if these carts are exactly the same weight. So the cart with the smaller wheels would need extra weight to compensate if they are to be an exact match. In that case, I believe the equal mass would cause them to travel the same distance. The smaller wheel would have more revolutions, and go faster, but the larger wheel, while moving slower would require less revolutions to make the same distance, and thus it balances out. If we're simply talking about 2 wheels side by side, Then yes, the small wheel with less mass will travel farther.

Staff Emeritus
Gold Member
Uh yes, of course points on the outside edge of a rotating body move faster than points near the center.

The velocity is called linear or tangential velocity, and it is described by:

v = w r

where w is the angular velocity (in radians/sec) and r is the distance from the axle.

- Warren