1. A system is composed of a rod and a disk, as shown at t = 0 below: 6 -6 4 6 -2 -4 -6 The axes have units of meters and both objects are initially stationary. The masses of the disk and rod are 10 kg and 8 kg, respectively. The following forces act on the center of the disk: N Fai - [24] and F₁2-[N] = = N 5 and the following force acts on the top of the rod: = - [2N] (a) Find system's initial center-of-mass (COM) position, com (0). (b) What is com (2 s)? -4 -2 4 2 2
1. A system is composed of a rod and a disk, as shown at t = 0 below: 6 -6 4 6 -2 -4 -6 The axes have units of meters and both objects are initially stationary. The masses of the disk and rod are 10 kg and 8 kg, respectively. The following forces act on the center of the disk: N Fai - [24] and F₁2-[N] = = N 5 and the following force acts on the top of the rod: = - [2N] (a) Find system's initial center-of-mass (COM) position, com (0). (b) What is com (2 s)? -4 -2 4 2 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
A diagram is provided with a grid background (units in meters) showing the positions of a red rod and red disk. The rod is positioned vertically on the left, spanning from coordinates (-6, -1) to (-6, 1). The disk is positioned on the right side around coordinates (4, 4).
**Given Data:**
The masses of the disk and rod are 10 kg and 8 kg, respectively.
The forces acting on the center of the disk are:
\[ \vec{F}_{d1} = \begin{pmatrix} 2 \, \text{N} \\ -4 \, \text{N} \end{pmatrix} \quad \text{and} \quad \vec{F}_{d2} = \begin{pmatrix} 0 \, \text{N} \\ 5 \, \text{N} \end{pmatrix} \]
The force acting on the top of the rod is:
\[ \vec{F}_{r} = \begin{pmatrix} 2 \, \text{N} \\ 2 \, \text{N} \end{pmatrix} \]
**Tasks:**
(a) Find the system’s initial center-of-mass (COM) position, \( \vec{r}_{\text{com}}(0) \).
(b) What is \( \vec{r}_{\text{com}}(2 \, \text{s}) \)?
### Diagrams/Graphs Explanation:
The provided diagram or graph showcases a Cartesian coordinate system overlaid with a labeled grid in meters. The two primary objects, a rod and a disk, are marked as solid red shapes. The rod is aligned vertically on the left side of the graph, while the disk is situated on the right side. The exact coordinates for initial placement of these objects are visualized within the grid framework to aid in comprehension and calculations.
### Questions and Analysis:
(a) **Initial Center-of-Mass (COM) Calculation:**
The COM for a system can be determined using the position coordinates of both components weighted by their masses.
\[ \vec{r}_{\text{com}} = \frac{m_{d}\vec{r}_{d} + m_{r](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F853bd72d-1b9a-40ff-a061-0c00f168c5bc%2F1bffd65b-0ae1-4780-8646-6a54a1ddbed0%2Fa1onpa_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
**1. A system is composed of a rod and a disk, as shown at \( t = 0 \) below:**
A diagram is provided with a grid background (units in meters) showing the positions of a red rod and red disk. The rod is positioned vertically on the left, spanning from coordinates (-6, -1) to (-6, 1). The disk is positioned on the right side around coordinates (4, 4).
**Given Data:**
The masses of the disk and rod are 10 kg and 8 kg, respectively.
The forces acting on the center of the disk are:
\[ \vec{F}_{d1} = \begin{pmatrix} 2 \, \text{N} \\ -4 \, \text{N} \end{pmatrix} \quad \text{and} \quad \vec{F}_{d2} = \begin{pmatrix} 0 \, \text{N} \\ 5 \, \text{N} \end{pmatrix} \]
The force acting on the top of the rod is:
\[ \vec{F}_{r} = \begin{pmatrix} 2 \, \text{N} \\ 2 \, \text{N} \end{pmatrix} \]
**Tasks:**
(a) Find the system’s initial center-of-mass (COM) position, \( \vec{r}_{\text{com}}(0) \).
(b) What is \( \vec{r}_{\text{com}}(2 \, \text{s}) \)?
### Diagrams/Graphs Explanation:
The provided diagram or graph showcases a Cartesian coordinate system overlaid with a labeled grid in meters. The two primary objects, a rod and a disk, are marked as solid red shapes. The rod is aligned vertically on the left side of the graph, while the disk is situated on the right side. The exact coordinates for initial placement of these objects are visualized within the grid framework to aid in comprehension and calculations.
### Questions and Analysis:
(a) **Initial Center-of-Mass (COM) Calculation:**
The COM for a system can be determined using the position coordinates of both components weighted by their masses.
\[ \vec{r}_{\text{com}} = \frac{m_{d}\vec{r}_{d} + m_{r
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