### Problem DescriptionA rigid, uniform, horizontal bar of mass \( m_1 \) and length \( L \) is supported by two identical massless strings. Both strings are vertical. String A is attached at a distance \( d < \frac{L}{2} \) from the left end of the bar and is connected to the ceiling, while String B is attached to the left end of the bar and is connected to the floor. A small block of mass \( m_2 \) is supported against gravity by the bar at a distance \( x \) from the left end of the bar, as shown in the figure.Throughout this problem, positive torque is that which spins an object counterclockwise. Use \( g \) for the magnitude of the free-fall acceleration due to gravity.### Tasks#### Part AFind \( T_A \), the tension in string A.- **Express the tension in string A in terms of** \( g, m_1, L, d, m_2, \) **and** \( x \).\[ T_A = \]#### Part BFind \( T_B \), the magnitude of the tension in string B.- **Express the magnitude of the tension in string B in terms of** \( T_A, m_1, m_2, \) **and** \( g \).\[ T_B = \]#### Part CComplete previous part(s).#### Part DIf the mass of the block is too large and the block is too close to the left end of the bar (near string B), then the horizontal bar may become unstable (i.e., the bar may no longer remain horizontal).- **What is the smallest possible value of \( x \) such that the bar remains stable (call it** \( x_{\text{critical}} \))?**- **Express your answer for** \( x_{\text{critical}} \) **in terms of** \( m_2, m_1, d, \) **and** \( L \).\[ x_{\text{critical}} = \]### Diagram ExplanationThe diagram illustrates a horizontal bar supported by two vertical strings. String A is positioned at a distance \( d \) from the left end, while String B is attached at the very end. A block is placed on the bar at a distance \( x \) from the left, contributing to tension and torque forces in the system.

Equilibrium of moment :∑ M=0m2gx+m1gL2+-TA×d=0TA=gd[m1L2+m1x]bEquilibrium of force along y :∑ Fy=0TA+TB-m2g-m1g=0TB=TA-(m1+m2)gdMBLOCK+MBAR=0m2gd-x+m1gd-L2=0xc=m1m2(d-L2)+d

Engineering

Civil Engineering

fof maee meis supported against gravity by the bar at a A rigid, uniform, horizontal bar of mass m, and length L is supported by two identical massless strings (Figure 1)Both strings are vertical. String A is attached at a distance d

fof maee meis supported against gravity by the bar at a A rigid, uniform, horizontal bar of mass m, and length L is supported by two identical massless strings (Figure 1)Both strings are vertical. String A is attached at a distance d

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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Concept explainers

System of Forces

A force is an external influence or agent that changes the state of a system. According to Newton's second law, the force applied to a body is proportional to the change in momentum of a body. Hence, whenever a force is applied to a body, its momentum changes. If the body is at rest, under the action of force, its state of rest changes to motion. In other words, the magnitude of the momentum of the body changes from zero to a measurable value. Similarly, if the body possesses some initial velocity, it has some associated momentum, the effect of force can cause the body to halt or come to rest, that is, the momentum reduces to zero.

Question

### Problem Description

A rigid, uniform, horizontal bar of mass \( m_1 \) and length \( L \) is supported by two identical massless strings. Both strings are vertical. String A is attached at a distance \( d < \frac{L}{2} \) from the left end of the bar and is connected to the ceiling, while String B is attached to the left end of the bar and is connected to the floor. A small block of mass \( m_2 \) is supported against gravity by the bar at a distance \( x \) from the left end of the bar, as shown in the figure.

Throughout this problem, positive torque is that which spins an object counterclockwise. Use \( g \) for the magnitude of the free-fall acceleration due to gravity.

### Tasks

#### Part A
Find \( T_A \), the tension in string A.

- **Express the tension in string A in terms of** \( g, m_1, L, d, m_2, \) **and** \( x \).

\[ T_A = \]

#### Part B
Find \( T_B \), the magnitude of the tension in string B.

- **Express the magnitude of the tension in string B in terms of** \( T_A, m_1, m_2, \) **and** \( g \).

\[ T_B = \]

#### Part C
Complete previous part(s).

#### Part D
If the mass of the block is too large and the block is too close to the left end of the bar (near string B), then the horizontal bar may become unstable (i.e., the bar may no longer remain horizontal).

- **What is the smallest possible value of \( x \) such that the bar remains stable (call it** \( x_{\text{critical}} \))?**
- **Express your answer for** \( x_{\text{critical}} \) **in terms of** \( m_2, m_1, d, \) **and** \( L \).

\[ x_{\text{critical}} = \]

### Diagram Explanation

The diagram illustrates a horizontal bar supported by two vertical strings. String A is positioned at a distance \( d \) from the left end, while String B is attached at the very end. A block is placed on the bar at a distance \( x \) from the left, contributing to tension and torque forces in the system.

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