A wheel that is initially at rest is mounted on an axle that is not frictionless. When a constant external torque of 50.0 N·m is applied to the wheel for 25.0 s the wheel angularly accelerates to an angular speed of 750 rpm. The external torque is removed and the wheel comes to rest after 120 s due to the friction at the axle. Write down N2L for each situation. What is (a) the moment of inertia of the wheel and (b) the magnitude of the frictional torque on the wheel at the axle (which is assumed to be constant)? (c) Include a complete diagram of each situation.
Angular Momentum
The momentum of an object is given by multiplying its mass and velocity. Momentum is a property of any object that moves with mass. The only difference between angular momentum and linear momentum is that angular momentum deals with moving or spinning objects. A moving particle's linear momentum can be thought of as a measure of its linear motion. The force is proportional to the rate of change of linear momentum. Angular momentum is always directly proportional to mass. In rotational motion, the concept of angular momentum is often used. Since it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics. To understand the concept of angular momentum first we need to understand a rigid body and its movement, a position vector that is used to specify the position of particles in space. A rigid body possesses motion it may be linear or rotational. Rotational motion plays important role in angular momentum.
Moment of a Force
The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.
![**Problem Description:**
A wheel that is initially at rest is mounted on an axle that is not frictionless. When a constant external torque of 50.0 N·m is applied to the wheel for 25.0 s, the wheel angularly accelerates to an angular speed of 750 rpm. The external torque is removed and the wheel comes to rest after 120 s due to the friction at the axle.
**Tasks:**
1. **Apply Newton’s Second Law (N2L) for Angular Motion**:
- For each situation, write down Newton’s Second Law (N2L) which is given by:
\[
\tau = I \alpha
\]
where \(\tau\) is the torque, \(I\) is the moment of inertia, and \(\alpha\) is the angular acceleration.
2. **Calculate the Moment of Inertia (I) of the Wheel**:
- Use the given torque and conditions to find the moment of inertia.
3. **Determine the Magnitude of the Frictional Torque**:
- Calculate the constant torque due to friction required to bring the wheel to rest.
4. **Create a Diagram for Each Situation**:
- Include a diagram representing the forces and motion of the wheel during the application of external torque and when frictional torque acts.
**Explanations (optional for visual content):**
- **Graph/Diagram Description**:
- Include diagrams showing:
- The first phase where external torque is applied, illustrating the acceleration.
- The second phase with only frictional torque acting to decelerate the wheel to rest.
Make sure to relate the calculations to the diagrams if they are used.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4dca08d-d7c1-4511-81d7-e3b17171b463%2Fd9a74bf3-d5ec-4250-bb42-3608d9a60243%2F1d9plrp_processed.png&w=3840&q=75)
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