91Ó°ÊÓ

Torque is a measure of how much a force acting on an object causes that object to rotate. In this case, the force applied to the bicycle wheel generates a torque of 5.00 N·m, leading to the rotational motion of the wheel. Torque can be thought of as the rotational equivalent of linear force. It's calculated as the product of the force applied and the distance from the rotation axis, known as the lever arm.
The formula for torque ( au) is:

• Torque ( \(\tau\)) = Force (F) × Lever Arm (\(r\))

In our exercise, the torque applied to the wheel initiates its angular acceleration and ultimately changes its angular velocity over time.

The moment of inertia ( I), sometimes referred to as rotational inertia, is a measure of an object's resistance to changes in its rotation. It depends on the mass distribution relative to the rotation axis and plays a key role in determining how much torque is needed for a desired angular acceleration.
For the bicycle wheel, the moment of inertia is calculated using the known torque and the angular acceleration derived during motion:

• \( \tau = I \alpha \rightarrow I = \frac{\tau}{\alpha} \)

In the given scenario, the moment of inertia ( I) is found to be approximately \( \frac{15}{\pi} \text{ kgâ‹…m}^2 \). This value indicates how the mass is distributed relative to the center of the wheel, enhancing understanding of the wheel's behavior under the applied torque.

Angular acceleration ( \alpha) describes how quickly an object's angular velocity changes over time. It's the rotational analog to linear acceleration, showing how fast an object speeds up or slows down its rotation.
The step-by-step solution determines this by finding the change in angular velocity over the time period in which the torque is applied. The formula is:

• \( \alpha = \frac{\Delta \omega}{\Delta t} \)

For the bicycle wheel, the angular acceleration is calculated to be \( \frac{5\pi}{3} \text{ rad/s}^2 \). This indicates the wheel's rate of increase in rotation speed during the 2 seconds of torque application.

Friction is the resistive force that slows down motion and, in this case, it's responsible for eventually stopping the wheel's rotation. As the external torque ceases, the wheel's motion is halted only by the friction in its bearings.
Friction torque ( \tau_f) can be determined using the known moment of inertia and the angular deceleration as the wheel comes to rest:

• \( \tau_f = I \alpha_f \)

The computed friction torque of -0.4 N·m reveals the magnitude of resistance overcoming the wheel's inertia, reinforcing the practical understanding of friction's pivotal role in rotational dynamics.

Rotational kinematics is the study of the motion of objects in rotation, analogous to linear kinematics. It involves parameters such as angular displacement, angular velocity, and angular acceleration.
In the exercise, once the torque stops, the wheel undergoes a deceleration due solely to friction. The angular displacement ( \theta) during this period gives insight into how far the wheel rotates before coming to rest.
The formula used:

• \( \theta = \omega_0 t + \frac{1}{2} \alpha_f t^2 \)

This calculation shows that the wheel makes roughly 11.1 revolutions during the deceleration phase, encapsulating the essence of rotational kinematics by translating initial motion into tangible distance covered.