Consider a flatbed truck containing a crane modeled as a long thin rod pivoted at its end. The rod is resting on the cab of the truck as shown while the truck is moving at speed v and suddenly accelerates forward with magnitude a = 5m/s². Assume the rod is L = 2 m long, weights m = 50 kg, and leans at an angle 0 = 60 deg. as shown. ING/O= (v- ¹aG/O = (a Öl sin 0 01 2 Part A: Assuming the inertial reference frame oriented as shown, verify (by showing all interme- diate steps) that the inertial velocity and acceleration of the center of mass point G are: 0²1 2 B O' •B bG sin 0) ²₁+ ( 01 cos 0) 12 Öl cos) ₁ + (cos v a Ö = (Nx sine - Ny cos 0 + N₂ sin 0) 6 ml 8²1 2 sin 0) 2₂ Part B: Using Euler's 2nd Law, show that the equation of motion for the angle of the crane is where N is the magnitude of the horizontal reaction force at B and Rx, Ry are the components of the reaction force at the pin O'. Part C: Consider Euler's 1st Law, along with your previous results, show that the normal force at B is 16.60 N if the crane is at rest ( = 8 = 0). Part D: If the acceleration is sufficiently large, the normal force at B goes to zero and the crane will begin to tip away from the wall of the truck. Show that the acceleration at which this occurs is amax = 5.66 m/s².
Consider a flatbed truck containing a crane modeled as a long thin rod pivoted at its end. The rod is resting on the cab of the truck as shown while the truck is moving at speed v and suddenly accelerates forward with magnitude a = 5m/s². Assume the rod is L = 2 m long, weights m = 50 kg, and leans at an angle 0 = 60 deg. as shown. ING/O= (v- ¹aG/O = (a Öl sin 0 01 2 Part A: Assuming the inertial reference frame oriented as shown, verify (by showing all interme- diate steps) that the inertial velocity and acceleration of the center of mass point G are: 0²1 2 B O' •B bG sin 0) ²₁+ ( 01 cos 0) 12 Öl cos) ₁ + (cos v a Ö = (Nx sine - Ny cos 0 + N₂ sin 0) 6 ml 8²1 2 sin 0) 2₂ Part B: Using Euler's 2nd Law, show that the equation of motion for the angle of the crane is where N is the magnitude of the horizontal reaction force at B and Rx, Ry are the components of the reaction force at the pin O'. Part C: Consider Euler's 1st Law, along with your previous results, show that the normal force at B is 16.60 N if the crane is at rest ( = 8 = 0). Part D: If the acceleration is sufficiently large, the normal force at B goes to zero and the crane will begin to tip away from the wall of the truck. Show that the acceleration at which this occurs is amax = 5.66 m/s².
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![**Description of Problem:**
Consider a flatbed truck containing a crane modeled as a long thin rod pivoted at its end. The rod is resting on the cab of the truck as shown while the truck is moving at speed \( v \) and suddenly accelerates forward with magnitude \( a = 5 \, \text{m/s}^2 \). Assume the rod is \( L = 2 \, \text{m} \) long, weighs \( m = 50 \, \text{kg} \), and leans at an angle \( \theta = 60^\circ \), as shown.
**Diagram Explanation:**
The image contains two diagrams:
1. A side view of a truck with a crane modeled as a rod leaning on the truck's cab. It shows the angle \( \theta \) between the rod and the horizontal, and points \( B \), \( G \), and \( O \) with vectors and labels indicating forces and dimensions.
2. A detailed view of the rod illustrating its orientation with respect to the inertial and body-fixed coordinate systems. The axes are labeled \( i_1, i_2, i_3 \) for the inertial frame and \( b_1, b_2, b_3 \) for the body-fixed frame.
---
**Part A:**
Assuming the inertial reference frame oriented as shown, verify (by showing all intermediate steps) that the inertial velocity and acceleration of the center of mass point \( G \) are:
\[
^I \mathbf{v}_{G/O} = \left(v - \frac{\dot{\theta} l}{2} \sin \theta \right) \mathbf{i}_1 + \left( \frac{\dot{\theta} l}{2} \cos \theta \right) \mathbf{i}_2
\]
\[
^I \mathbf{a}_{G/O} = \left(a - \frac{\ddot{\theta} l}{2} \sin \theta - \frac{\dot{\theta}^2 l}{2} \cos \theta \right) \mathbf{i}_1 + \left( \frac{\ddot{\theta} l}{2} \cos \theta - \frac{\dot{\theta}^2 l}{2} \sin \theta \right) \mathbf{i}_2
\]
---
**Part B:**
Using Euler’s](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd23cf25f-e0e1-420e-8dcb-8ea8662c7deb%2F16cf1855-dc8b-44af-8e62-5e93116fb3e8%2Fl8nsqf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Description of Problem:**
Consider a flatbed truck containing a crane modeled as a long thin rod pivoted at its end. The rod is resting on the cab of the truck as shown while the truck is moving at speed \( v \) and suddenly accelerates forward with magnitude \( a = 5 \, \text{m/s}^2 \). Assume the rod is \( L = 2 \, \text{m} \) long, weighs \( m = 50 \, \text{kg} \), and leans at an angle \( \theta = 60^\circ \), as shown.
**Diagram Explanation:**
The image contains two diagrams:
1. A side view of a truck with a crane modeled as a rod leaning on the truck's cab. It shows the angle \( \theta \) between the rod and the horizontal, and points \( B \), \( G \), and \( O \) with vectors and labels indicating forces and dimensions.
2. A detailed view of the rod illustrating its orientation with respect to the inertial and body-fixed coordinate systems. The axes are labeled \( i_1, i_2, i_3 \) for the inertial frame and \( b_1, b_2, b_3 \) for the body-fixed frame.
---
**Part A:**
Assuming the inertial reference frame oriented as shown, verify (by showing all intermediate steps) that the inertial velocity and acceleration of the center of mass point \( G \) are:
\[
^I \mathbf{v}_{G/O} = \left(v - \frac{\dot{\theta} l}{2} \sin \theta \right) \mathbf{i}_1 + \left( \frac{\dot{\theta} l}{2} \cos \theta \right) \mathbf{i}_2
\]
\[
^I \mathbf{a}_{G/O} = \left(a - \frac{\ddot{\theta} l}{2} \sin \theta - \frac{\dot{\theta}^2 l}{2} \cos \theta \right) \mathbf{i}_1 + \left( \frac{\ddot{\theta} l}{2} \cos \theta - \frac{\dot{\theta}^2 l}{2} \sin \theta \right) \mathbf{i}_2
\]
---
**Part B:**
Using Euler’s
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