Atte Consider the objects labeled A, B, C, and D shown in the figure. 米1- B C D Each object is composed of identical thin sticks of uniformly distributed mass 9.15 kg and length 0.489 m. What are the moments of inertia IA, IB, Ic, and Ip of the objects about a rotation axis perpendicular to the screen and passing through the black dot displayed on each object?
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.
![### Interactive Exercise on Calculating Moments of Inertia
**Objective:**
Calculate the moments of inertia for different objects composed of identical thin sticks using the provided diagram.
**Diagram Explanation:**
The figure displays four objects, labeled A, B, C, and D.
- **Object A**: Formed by arranging six thin sticks in a star pattern, all intersecting at a central black dot.
- **Object B**: Consists of two perpendicular sticks intersecting at a central black dot.
- **Object C**: Four sticks arranged in a straight line with a black dot midpoint.
- **Object D**: Consists of three sticks, two aligned horizontally and one vertically, meeting at a central black dot.
**Description:**
Each object is composed of identical thin sticks with the following properties:
- Mass: 9.15 kg
- Length: 0.489 m
**Task:**
Determine the moments of inertia \( I_A, I_B, I_C, \) and \( I_D \) for each object. The rotation axis is perpendicular to the screen and passes through the black dot on each object.
**Calculations:**
- \( I_A = 0.702 \, \text{kg} \cdot \text{m}^2 \) — Incorrect
- \( I_B = 0.8784 \, \text{kg} \cdot \text{m}^2 \) — Incorrect
- \( I_C = 5.62 \, \text{kg} \cdot \text{m}^2 \)
- \( I_D = 5.79 \, \text{kg} \cdot \text{m}^2 \)
Complete the calculations and verify your values for \( I_A \) and \( I_B \). Use the given formula for moment of inertia for thin rods rotated about an axis:
\[
I = \frac{1}{12} m L^2 + m d^2
\]
Where \( m \) is the mass, \( L \) is the length, and \( d \) is the distance to the axis of rotation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b7ec4e2-929f-4e5a-b9e2-d730b007ea8c%2F9b423597-926d-467a-80c9-f27932c433dd%2Fsuplnnf_processed.jpeg&w=3840&q=75)
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