The wheels of a wagon can be approximated as the combination of a thin outer hoop, of radius ?h=0.580 m and mass 4.51 kg, and two thin crossed rods of mass 8.66 kg each. A farmer would like to replace his wheels with uniform disks ?d=0.0651 m thick, made out of a material with a density of 5990 kg per cubic meter. If the new wheel is to have the same moment of inertia about its center as the old wheel about its center, what should the radius of the disk be?
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.
![### Moment of Inertia of Common Shapes
Understanding the moment of inertia is crucial in the study of rotational dynamics. Below, we illustrate and explain the moment of inertia for a wheel and a disk.
#### Diagram Explanation
The diagram consists of two parts:
1. **Wheel with Spokes:**
- The first image on the left shows a wheel with four spokes.
- The radius of the wheel is denoted as \( r_h \).
2. **Solid Disk:**
- The second image on the right shows a solid disk.
- The disk's radius is denoted as \( r_d \).
- The disk's thickness is denoted as \( t_d \).
#### Moment of Inertia Formulas
- **Wheel with Spokes:**
The moment of inertia (\( I \)) of a wheel, especially one with spokes like the one shown, can be calculated using:
\[ I = M r_h^2 \]
where \( M \) is the mass of the wheel and \( r_h \) is the radius of the wheel.
- **Solid Disk:**
The moment of inertia (\( I \)) of a solid disk is given by:
\[ I = \frac{1}{2} M r_d^2 \]
where \( M \) is the mass of the disk and \( r_d \) is the radius of the disk.
#### Applications
Understanding these fundamental shapes' moments of inertia is pivotal in various real-world applications, such as engineering, mechanical design, and physics.
#### Summary
- **Wheel:** Moment of inertia depends mainly on the radius and takes into account the mass concentrated around the perimeter.
- **Disk:** With evenly distributed mass, the disk’s moment of inertia considers both the radius and the mass.
This topic lays the groundwork for more advanced studies in rotational motion and dynamics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5039f9f1-9031-4cf0-b840-7355f91f2cd2%2Fe354112c-456c-4e6d-a668-a915d8f1e9b9%2Fk1j2vkm.jpeg&w=3840&q=75)
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