# Spinning bicycle wheel in air and letting it go

• I
• ChessEnthusiast
In summary: The final velocity of the wheel will depend on the relative magnitude of the tangential acceleration and the angular acceleration.In summary, the problem involves a bicycle wheel with a radius of 25 cm and a mass of 1 kg, spinning in air at a speed of 50 km/h. When placed on asphalt, the wheel will experience kinetic friction, which will reduce its speed. The final velocity of the wheel will depend on the balance between its tangential and angular acceleration. This can be solved using the momentum-impulse rule and the relationship between terminal velocity and angular velocity.

#### ChessEnthusiast

Let's say that we have a bicycle wheel with radius $R = 25 cm$ and mass $m = 1 kg$. We spin the wheel in air so, that the tread is moving at the speed of $v_o = 50 km / h$.
Then, we let it roll on asphalt, given the coefficient of static friction: $f_s = 0.9$ and kinetic friction: $f_k = 0.7$. (I am making these numbers up, I hope they work)
I wonder what the final velocity of the wheel will be (after traction is "restored").

This is how I attempted to solve this problem:

Once this wheel is put on asphalt, it will start moving with the initial speed of $v_o$. Then, the force of kinetic friction $F = f_k mg$ will reduce its speed by $\Delta v$. By the momentum-impulse rule,
$$m \Delta v = F \Delta t$$

The torque generated by the force of kinetic friction will affect the angular velocity like this:
$$f_k mg \cdot R =I \frac{\Delta \omega}{\Delta t}$$

Now, using the fact that once the terminal velocity is reached, the wheel will no longer skid - this relation holds
$$v_{terminal} = \omega_{terminal} \cdot R$$

Now, this system of equations is actually solvable, but I am not sure whether my reasoning is correct. Any ideas?

Similar problems have been done here before, based on spheres, and possibly hollow cylinders, which would be the same as a bicycle wheel. The initial linear speed of the wheel will be zero. The wheel then accelerates linearly due to the kinetic friction which also decreases the wheel's angular velocity. Eventually the wheel will transition into pure rolling.

## What is the concept behind a spinning bicycle wheel in air?

The concept behind a spinning bicycle wheel in air is related to the conservation of angular momentum. When the wheel is spinning, it has angular momentum which is a measure of its rotational motion. According to the law of conservation of angular momentum, this momentum must remain constant unless an external force is applied.

## Why does the bicycle wheel spin faster when held by the handles?

When the bicycle wheel is held by the handles, the force of gravity is acting on the weight of the wheel. This weight creates a torque, or rotational force, which causes the wheel to spin faster. This is due to the fact that the distance from the axis of rotation (the center of the wheel) to the point where the force is applied (the handles) is greater, creating a larger torque.

## What happens when the spinning bicycle wheel is released?

When the spinning bicycle wheel is released, the force of gravity acting on the weight of the wheel causes it to start rotating around a different axis. This is known as precession and is a result of the conservation of angular momentum. As the wheel precesses, the handlebars will also move in a circular motion.

## Why does the direction of precession change when the spinning wheel is tilted?

The direction of precession changes when the spinning wheel is tilted because the point of contact between the wheel and the ground changes. This changes the direction of the torque acting on the wheel, causing it to precess in a different direction. This phenomenon is known as gyroscopic precession.

## Can the spinning bicycle wheel continue to precess indefinitely?

No, the spinning bicycle wheel will eventually slow down and stop precessing due to frictional forces. These forces act to slow down the rotation of the wheel and eventually bring it to a stop. However, with minimal friction, the wheel can precess for a longer period of time.