Question

A bicycle wheel of mass M (assumed to be concentrated at its rim) and radius R is mounted horizontally so it may turn without friction on a vertical axle. A dart of mass mo is thrown with velocity vo as shown above and sticks in the tire.

 a. If the wheel is initially at rest, find its angular velocity co after the dart strikes

 b. In terms of the given quantities, determine the ratio:

 final kinetic energy of the system

 initial kinetic energy of the system

Answer 1

Answer 2

The mass of a bicycle wheel is concentrated at the rim for several reasons, primarily to enhance its rotational inertia and stability while in motion. Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to changes in its rotation. In the case of a bicycle wheel, having most of the mass concentrated at the rim affects its rotational inertia in significant ways:

1. Increased Rotational Inertia: The farther the mass is from the center of rotation (the hub of the wheel, in this case), the higher the rotational inertia. By placing mass at the rim, the rotational inertia of the wheel increases. This means that it takes more force to change the rotational motion of the wheel, providing stability when the bicycle is in motion. It's similar to how it's easier to spin with your arms close to your body and harder when your arms are extended.

2. Smoothing out Motion: A wheel with more mass at the rim has more momentum, which helps to smooth out the motion of the bicycle. When you encounter bumps or irregularities on the road, the extra momentum helps the wheel resist sudden changes in motion, leading to a more stable ride.

3. Efficient Energy Transfer: When pedaling, the mass at the rim assists in the efficient transfer of energy. The energy from your legs is transferred to the pedals, and eventually to the rim due to the chain and gear system. Having more mass at the rim means that the energy is distributed effectively throughout the wheel, making the ride more efficient.

4. Gyroscopic Effect: The rotating wheel generates a gyroscopic effect, which provides stability by resisting changes in the direction of motion. This effect is more pronounced when the mass is concentrated at the rim, enhancing the overall stability of the bicycle.

In summary, concentrating the mass at the rim of a bicycle wheel enhances its rotational inertia, stability, and efficiency, making the bicycle easier to control and ride, especially at higher speeds or in varied terrains.