Question

With center and spokes of negligible mass, a certain bicycle wheel has a thin rim of radius \(36.3 \mathrm{~cm}\) and mass \(3.66 \mathrm{~kg}\); it can turn on its axle with negligible friction. A man holds the wheel above his head with the axis vertical while he stands on a turntable free to rotate without friction; the wheel rotates clockwise, as seen from above, with an angular speed of \(57.7 \mathrm{rad} / \mathrm{s}\), and the turntable is initially at rest. The rotational inertia of wheel-plus-man-plus-turntable about the common axis of rotation is \(2.88 \mathrm{~kg} \cdot \mathrm{m}^{2} .(a)\) The man's hand suddenly stops the rotation of the wheel (relative to the turntable). Determine the resulting angular velocity (magnitude and direction) of the system. (b) The experiment is repeated with noticeable friction introduced into the axle of the wheel, which, starting from the same initial angular speed \((57.7 \mathrm{rad} / \mathrm{s})\), gradually comes to rest (relative to the turntable) while the man holds the wheel as described above. (The turntable is still free to rotate without friction.) Describe what happens to the system, giving as much quantitative information as the data permit.

Short answer

The resulting angular velocity of the system is roughly 9.7 rad/s clockwise. When friction is introduced to the wheel's axle and makes the wheel come to a stop, the rest of the system gains angular momentum to conserve the total angular momentum of the entire system.

Step-by-step solution

Calculate Initial Angular Momentum

Firstly, we will calculate the angular momentum of the wheel before it is stopped. The angular momentum of a rotating object is given by \(L = Iω \), where \(L\) is the angular momentum, \(I\) is the moment of inertia, and \(ω\) is the angular velocity. The moment of inertia \(I\) of a thin rim is given by \(I = mR^2\), where \(m\) is the mass of the rim, and \(R\) is the radius. Substituting these values, we obtain \(I = (3.66 kg) * (0.363 m)^2 \approx 0.483 kgm^2\). Hence, the initial angular momentum (\(L_i\)) of only the wheel is \(L_i = Iω = (0.483 kgm^2)(57.7 rad/s) \approx 27.9 kgm^2/s\).

Calculate Final Angular Momentum

After the wheel is stopped, according to the law of conservation of angular momentum, the angular momentum of the system (wheel + man + turntable) will remain same. The final moment of inertia is given in the problem as \(2.88 kgm^2\) and final angular velocity is 'ωf'. Therefore, Final angular momentum (\(L_f\)) = \(2.88 kgm^2 * ωf\). But, we know that \(L_f = L_i\) (From law of conservation of angular momentum), so, we can equate the two and solve for \(ωf\) = \(L_i / 2.88 kgm^2 = 27.9 kgm^2/s / 2.88 kgm^2 \approx 9.7 rad/s\)

Explain Direction of Final Angular Velocity

Now, since the wheel was rotating clockwise initially and no external torques have been applied, the final angular velocity will also be in the clockwise direction according to conservation of angular momentum.

Anticipate Impact of Friction

When there is friction in the axle of the wheel, the wheel will gradually lose its angular momentum as it comes to rest. As the angular momentum of the wheel decreases due to friction, the angular momentum of the rest of the system (man + turntable) would increase to conserve the total angular momentum. Due to increased friction, the man, wheel, and turntable will all rotate together, until the wheel loses all of its angular momentum.

Key concepts

Angular Momentum Conservation

Angular momentum conservation is a principle that arises from Newton's laws of motion applied to rotating systems. It states that when there are no external torques acting on a system, its total angular momentum remains constant. This principle is crucial in understanding the exercise about the bicycle wheel and turntable.

In this scenario, the initial state has the bicycle wheel spinning with a certain angular momentum, while the man's resistance on the wheel results in an equal and opposite reaction so that the turntable and the man start spinning. Here's a deeper dive into the concept:

• **Initial State:** The wheel has an angular momentum calculated by the product of its moment of inertia and angular velocity. Initially, L = Iω, where I = 0.483 kg·m² and ω = 57.7 rad/s.
• **Final State:** After the wheel is stopped, the conservation principle dictates that the total angular momentum of the wheel plus turntable-man system is unchanged, even though the distribution between the wheel and the man+turntable changes.
• **Direction:** Since there's no external torque, the rotational direction is conserved. Initially clockwise, it remains clockwise as the action unfolds.

The problem highlights how angular momentum can be transferred within a system while remaining conserved as a whole.

Moment of Inertia

The moment of inertia is like the counterpart of mass in linear motion, indicating how much an object resists changes in its rotation around an axis. It is a key component in calculating rotational kinetic energy and angular momentum, thus essential in the exercise.

- **Bicycle Wheel:** For objects like a thin rim (representing the bicycle wheel), the moment of inertia can be determined by the formula \(I = mR^2\), where \(m\) is the mass and \(R\) is the radius.
- **System Moments:** While the wheel alone has a specific moment of inertia (calculated as 0.483 kg·m²), the scenario provides the total moment of inertia for the wheel-man-turntable system as 2.88 kg·m². This includes inertial effects from other components outside the wheel itself.

Understanding these values helps illustrate how the system shares its rotational motion and adjusts to conservational laws through its geometry and mass distribution. When the man stops the wheel, the turntable takes on additional rotation to maintain total angular momentum.

Torque and Angular Motion

Torque is a measure of how much a force causes an object to rotate. In the context of the exercise, we did not have any external torque applied after the initial setup, maintaining a natural state of motion without external interference.

- **Relation to Angular Motion:** Torque connects directly with angular motion through Newton’s Second Law for Rotation: \(\tau = I\alpha\), where \(\tau\) is the torque, \(I\) the moment of inertia, and \(\alpha\) the angular acceleration.
- **Friction Effects:** Initially, there was negligible friction. However, in the scenario exploring friction, gradual spinning down due to internal torques came into play as friction worked to slow the wheel. Friction introduces external forces that could affect angular momentum by changing the internal distribution of momentum among components.

This exercise demonstrates how torque influences whether and how the whole system speeds up or slows down without any direct external torques but by distributing existing angular forces through friction.