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Equilibrium of Forces Acting at a Point

Introduction

Addition of Forces

Forces are one of a group of quantities known as vectors, which are distinguished from regular numbers (know as scalars) by the fact that a vector has two quantities associated with it, a magnitude and a direction (related to a coordinate axes of the system you are dealing with). These properties completely characterize a vector. A vector may alternatively be described by specifying its vector components. In the case of the Cartesian coordinate system (the system we will be primarily dealing with) there are two components, the x-component and y-component. These two properties also completely characterize a vector. Vectors, and in the case of this lab, force vectors, can be represented pictorially (see Fig. 1) by an arrow pointing in the direction of action of the force, with a length proportional to the strength (magnitude) of the force.
Figure 1

Figure 1

The components
Fx 
and
Fy 
in the x and y directions of the vector F are related to the magnitude F and angle θ by:
( 1 )
Fx = F cos θ and Fy = F sin θ 
and conversely:
( 2 )
F =
Fx2 + Fy2
, and θ = arctan
Fy
Fx
.
 
When several forces act on a point, their sum can be obtained according to the rules of vector algebra. Graphically, the sum of two forces
F = F1 + F2 
can be found by using the parallelogram rule illustrated in Fig. 2 or, equivalently, by the head-to-tail method illustrated in Fig. 3.
Figure 2

Figure 2

Figure 3

Figure 3

The sum of the vectors can also be derived analytically by adding their components:
( 3 )
Fx = F1x + F2x, and Fy = F1y + F2y 

Condition for Translational Equilibrium

An object is in translational equilibrium when the vector sum of all the forces acting on it is zero. In this experiment we shall study the translational equilibrium of a small ring acted on by several forces on an apparatus known as a force table, see Fig. 4. This apparatus enables one to cause the forces of gravity acting on several masses (F = mg) to be brought to bear on the small ring. These forces are adjusted until equilibrium of the ring is achieved. You will then add the forces analytically by adding their components and graphically by drawing the vectors and determining if they add to zero using the rules for the addition of force vectors listed above.
Figure 4

Figure 4

Procedure

Equilibrium with Three Forces

We shall first study the equilibrium of the small ring when there are three forces acting on it. Two of the forces
(F1 and F2
will be fixed and the third one
F3 
adjusted until equilibrium is reached.

Equilibrium with Four Forces

Be sure to pledge your work, initial your data, and have your TA initial your data.

Analysis

Graphical Analysis

Make accurate diagrams on normal rectangular graph paper showing the sum of the forces acting on the ring for both parts of the experiment above.

Analytical Sum

Calculate the resultant force on the ring,
FT = F1 + F2 + F1
analytically for Equilibrium with Three Forces only. Choose zero degree to be the +x-axis, and 90° to be the +y-axis. The Analytical Sum section of the WebAssign question for this lab, will help facilitate the error analysis.

Discussion