Find the relationship between angles theta and phi using equations of equilibrium and solve for theta. Suppose that the ages are expressed in radians. EXPRESS YOUR EQUATION FOR THETA IN TERMS OF PHI. EXPRESS YOUR ANSWER IN RADIANS. I've show the equations for equilibrium in the attached image. ### Educational Content: Free-Body Diagram of a Point Particle**Learning Goal:** To draw the free-body diagram of a point particle, use the equations of equilibrium to find unknown forces, and understand how frictionless pulleys affect the transfer of force in a cable.As shown, a mass is suspended from a cable that wraps around a frictionless and massless pulley. The cable connects to a linear elastic spring. For this problem, treat the pulley as a point particle at \( B \).**Diagram Explanation (Figure 1):**This figure shows a mechanical setup involving a pulley system. Here is the detailed explanation of the components in the figure:- **Pulley (at point B):** The pulley is assumed to be frictionless and massless. It changes the direction of the tension force in the cable without adding any additional force components. - **Mass (hanging):** There is a mass hanging from the cable which is being subjected to gravitational forces.- **Cable:** The cable is inextensible, meaning its length does not change under tension.- **Spring:** The cable connects to a linear elastic spring, which exerts a force proportional to its extension or compression.- **Angles:** - \(\theta\): The angle between the horizontal direction and the force \(\overrightarrow{D}\) from the spring to the pulley. - \(\phi\): The angle between the vertical direction and the portion of the cable extending upwards from the pulley to the point where it's attached to the wall or ceiling.- **Coordinate System:** - The coordinate system is defined with \( x \) and \( y \) axes. The \( x \)-axis is horizontal, and the \( y \)-axis is vertical.The forces involved in the setup include:1. The tension in the cable.2. The elastic force exerted by the spring.3. The gravitational force acting on the mass.Understanding the interplay between these forces and the use of free-body diagrams will help in analyzing the system to determine unknown forces acting on the point particle at \( B \). ## Understanding the Relationship Between Angles θ and ϕ Using Equilibrium EquationsThis section explores the equilibrium of forces at a point in a system, represented by the given diagram and equations. We aim to find the relationship between the angles θ and ϕ. Note that all angles are expressed in radians.### Diagram ExplanationThe diagram illustrates a point A where three forces are acting. The forces are represented as vectors and their respective angles with the horizontal axis are indicated. The vectors are labeled as follows:- \( \vec{T} \) forms an angle ϕ with the horizontal.- \( \vec{T} \) forms an angle θ with the horizontal.- A third internal force at point A.### Analyzing Forces in Equilibrium To analyze the system, we ensure that the sum of forces in both the x-direction and y-direction equals zero, which is a fundamental condition for equilibrium.#### Equilibrium in the x-direction (ΣFx = 0):\[ T \cos ϕ - T \cos θ = 0 \]\[ T \cos ϕ = T \cos θ \]#### Equilibrium in the y-direction (ΣFy = 0):\[ T \sin θ - T_A + T \sin ϕ = 0 \]We are given the relationship \( T_A = T \).Using this, we substitute into the y-direction equation:\[ T \sin θ - T + T \sin ϕ = 0 \]\[ T \sin θ = T - T \sin ϕ \]\[ T \sin θ = T + T \sin ϕ \]### ConclusionBy solving these equilibrium equations, we can determine the relationship between angles θ and ϕ in the given system. The solved relationships provide insights into the force distribution and their dependency on the angles.### Detailed Steps:1. **Equilibrium in the x-direction**: \[ T \cos ϕ = T \cos θ \] Extract the relationship: \[ \cos ϕ = \cos θ \]2. **Equilibrium in the y-direction**: \[ T \sin θ - T + T \sin ϕ = 0 \] Simplifying, we get: \[ T \sin θ = T + T \sin ϕ \]This analytical approach helps in understanding the angles'

Consider the free body diagram of the figure as shown below, Since the pulley is mass-less and…

Answered: Find the relationship between angles…

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA