A force of magnitude P = 166 N is applied to the stationary machine handle as shown. Write the force and moment reactions at O as vectors. Neglect the weight of the handle assembly.
Plane Trusses
It is defined as, two or more elements like beams or any two or more force members, which when assembled together, behaves like a complete structure or as a single structure. They generally consist of two force member which means any component structure where the force is applied only at two points. The point of contact of joints of truss are known as nodes. They are generally made up of triangular patterns. Nodes are the points where all the external forces and the reactionary forces due to them act and shows whether the force is tensile or compressive. There are various characteristics of trusses and are characterized as Simple truss, planar truss or the Space Frame truss.
Equilibrium Equations
If a body is said to be at rest or moving with a uniform velocity, the body is in equilibrium condition. This means that all the forces are balanced in the body. It can be understood with the help of Newton's first law of motion which states that the resultant force on a system is null, where the system remains to be at rest or moves at uniform motion. It is when the rate of the forward reaction is equal to the rate of the backward reaction.
Force Systems
When a body comes in interaction with other bodies, they exert various forces on each other. Any system is under the influence of some kind of force. For example, laptop kept on table exerts force on the table and table exerts equal force on it, hence the system is in balance or equilibrium. When two or more materials interact then more than one force act at a time, hence it is called as force systems.
![### Problem Statement:
A force of magnitude \( P = 166 \, \text{N} \) is applied to the stationary machine handle as shown. Write the force and moment reactions at \( O \) as vectors. Neglect the weight of the handle assembly.
### Diagram Explanation:
The diagram shows a lever handle fixed at point \( O \), with a force \( P \) applied at point \( A \). The force \( P \) is directed at a 12° angle from the horizontal plane. The lever extends horizontally in the \( y \)-direction for 165 mm and vertically in the \( z \)-direction for 125 mm.
- **Coordinates:**
- \( x \)-axis: Horizontal to the left.
- \( y \)-axis: Horizontal, along the lever's main direction.
- \( z \)-axis: Vertical.
### Given Dimensions:
- Horizontal length along the \( y \)-axis: 165 mm
- Vertical height along the \( z \)-axis: 125 mm
### Answers:
**Force Reaction \( \mathbf{R} \) at \( O \):**
\[
R = (-162.37 \, \mathbf{i} + 0 \, \mathbf{j} - 34.51 \, \mathbf{k}) \, \text{N}
\]
**Moment Reaction \( \mathbf{M} \) at \( O \):**
\[
M = (-5694.15 \, \mathbf{i} - 2029.25 \, \mathbf{j} + 26791.05 \, \mathbf{k}) \, \text{N}\cdot\text{m}
\]
### Explanation:
- The force reaction vector \( \mathbf{R} \) shows components along the \( x \), \( y \), and \( z \) axes indicating its effect at point \( O \).
- The moment reaction vector \( \mathbf{M} \) represents the tendency of the force to cause rotation about each of the three axes at point \( O \). The significant \( \mathbf{k} \) component suggests a major rotational effect in the vertical plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F65a069a6-add9-4096-9c7b-9e8946da3313%2F6abf6c82-ba43-4967-838d-95453c0fcfec%2Fqgdcke_processed.png&w=3840&q=75)
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