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With axle and spokes of negligible mass and a thin rim, a certain bicycle wheel has a radius of \(0.350 \mathrm{~m}\) and weighs \(37.0 \mathrm{~N} ;\) it can turn on its axle with negligible friction. A man holds the wheel above his head with the axle vertical while he stands on a turntable that is 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 \(+\mathrm{man}+\) turntable about the common axis of rotation is \(2.10 \mathrm{~kg} \cdot \mathrm{m}^{2}\). The man's free hand suddenly stops the rotation of the wheel (relative to the turntable). Determine the resulting (a) angular speed and (b) direction of rotation of the system.

Step by step solution

01

Determine Wheel's Moment of Inertia

The wheel weighs 37.0 N, so its mass is \( m = \frac{37.0}{9.81} \approx 3.77 \text{ kg} \). The moment of inertia for the wheel using \( I = m r^2 \) is: \[ I = (3.77 \text{ kg})(0.350 \text{ m})^2 = 0.462 \text{ kg} \cdot \text{m}^2 \]

02

Apply Conservation of Angular Momentum

The initial angular momentum of the system is due to the wheel's rotation:\[ L_i = I_{wheel} \omega_{wheel} + I_{system} \cdot 0 = (0.462 \text{ kg} \cdot \text{m}^2)(57.7 \text{ rad/s}) = 26.66 \text{ kg} \cdot \text{m}^2/s \] when the wheel is stopped, the final momentum will be transferred entirely to the system:\[ L_f = I_{system} \cdot \omega_f \]

03

Solve for Final Angular Speed of the System

Using conservation of angular momentum, set initial and final momentum equal: \[ 26.66 \text{ kg} \cdot \text{m}^2/s = 2.10 \text{ kg} \cdot \text{m}^2 \cdot \omega_f \] \[ \omega_f = \frac{26.66}{2.10} \approx 12.69 \text{ rad/s} \]

04

Determine Direction of Rotation

The wheel initially rotates clockwise (as seen from above). By stopping the wheel, the conservation dictates the turntable and the person will start to move in the same clockwise direction to conserve momentum.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Moment of Inertia

The concept of moment of inertia is central to understanding rotational dynamics. In simple terms, it describes an object's resistance to change in its rotational state. Analogous to mass in linear motion, the moment of inertia depends on how an object's mass is distributed relative to the axis of rotation. This exercise considers a bicycle wheel with negligible mass from its spokes and axle, essentially treating it as a thin rim.

Here's how the moment of inertia is calculated in this context:

• The mass of the wheel is determined from its weight: 37.0 N, which translates into a mass of approximately 3.77 kg (by dividing the weight by gravitational acceleration, 9.81 m/s").
• The formula used is:
  \[ I = m imes r^2 \]
  where \( m \) is the mass of the object and \( r \) is the radius.
• For this wheel, \( I = 3.77 \text{ kg} \times (0.350 \text{ m})^2 = 0.462 \text{ kg} \cdot \text{m}^2 \).

Understanding moment of inertia is crucial because it lets us predict how objects behave when they spin.

Rotational Dynamics

Rotational dynamics is a key concept when dealing with objects in rotation. Similar to linear dynamics, it involves understanding forces and their effects on motion, but in a rotational context.

This exercise focuses on the conservation of angular momentum, which is a major principle within rotational dynamics:

• Angular momentum, denoted as \( L \), is the rotational equivalent of linear momentum and is given by \( L = I \times \omega \), where \( I \) is the moment of inertia and \( \omega \) is the angular speed.
• The law of conservation of angular momentum states that if no external torques act on a system, the total angular momentum remains constant.
• In this case, the initial angular momentum comes from the rotating bicycle wheel, and when stopped, this momentum transfers to the rest of the system.

The ability to conceptualize rotational dynamics and angular momentum is vital in calculating how and why systems move as they do.

Angular Speed

Angular speed is a measure of how quickly an object rotates or spins around an axis, expressed in radians per second (\( \text{rad/s} \)). It is akin to linear speed but describes rotational motion.

In the exercise:

• The bicycle wheel initially rotates at an angular speed of 57.7 \( \text{rad/s} \).
• When the wheel is stopped by the person, the angular speed of the entire system changes due to the transfer of angular momentum.
• Using conservation principles, the final angular speed of the system becomes \( \omega_f = 12.69 \text{ rad/s} \).
• This decrease in angular speed demonstrates how momentum transfer alters the speed without any additional external forces involved.

Understanding angular speed helps us grasp how rotational speeds adjust when interacting with different components.

Momentum Transfer

Momentum transfer is an important phenomenon in physics, involving the movement of momentum from one part of a system to another. This concept is vital in scenarios involving collisions or when objects interact with each other.

In this context:

• When the rotating bicycle wheel is suddenly stopped by the man, its momentum isn't lost but rather transferred to the man plus the turntable system.
• Initially, all the angular momentum resides with the wheel. After it stops, momentum conservation dictates this momentum shifts to the rest of the system, causing it to rotate.
• This momentum transfer results in the system having a new angular speed of approximately 12.69 \( \text{rad/s} \).
• The rotational direction also remains clockwise, consistent with conservation laws.

Understanding how momentum transfer works helps explain the resultant motion of interacting bodies and systems in rotational scenarios.