Date _________

Addition and Resolution of Vectors

Equilibrium of a Particle

Overview

When a set of forces act on an object in such a way that the lines of action of the forces pass through a common point, the forces are described as concurrent forces. When these forces lie in the same geometric plane, the forces are also described as coplanar forces. A single G

G

equivalent force known as the resultant force FR may replace a set of concurrent forces F1 and G

F2 , as shown. This resultant force is obtained by a process of vector addition of the original force vectors and produces the same effect as

the combined effect produced by all the

original forces. Conversely, a set of

concurrent forces can be balanced exactly by a

single force that acts at the common point of

concurrence of the forces. Such a force is

G

known as the equilibrant FE of that set of

forces and it is equal in magnitude but acts in

exactly opposite direction to the resultant of

the set of forces. A particle is considered to be

in (static) equilibrium under the action of a set

of forces when the vector sum of all the forces

is zero.

In this laboratory experiment, the student

will be introduced to methods of addition of

vectors. Using a force table, the student will determine the magnitudes and directions of applied concurrent forces, find the resultant force of a given set of vectors, investigate the relationship between the resultant force and the equilibrant force of a given set of forces, and compute the rectangular x- and y-components of known forces and their resultant force. The force table apparatus used in this experiment has a horizontally mounted circular table and the rim of this table is calibrated in degrees, from 0° to 360°. Forces of any chosen magnitude can be applied to a central ring (placed around a central pin at the center of the circular table) at any preferred angle by means of strings passing over a pulley and attached to a weight hanger. Each pulley can be adjusted to any chosen position around the rim of the circular table. The force exerted on the central ring is due to the gravitational force acting on the total mass on the weight hanger (including the hanger mass). Adding or removing masses on the weight hanger changes the magnitude of a force vector while moving the position of the pulley changes the direction of the force vector.

©2003 Martin O. Okafor

-1-

rev.1/22/04

Name _____________________ Class ______________

Date _________

Theory

Certain physical quantities, known as vectors, can only be completely described in terms of their magnitude and direction. Those quantities that can be completely described only in terms G

of their magnitudes are called scalars. A vector R is represented symbolically as R and G

graphically as an arrow (drawn to scale on graph paper). The length of the R

arrow is proportional to the magnitude of the vector (R) while the tip of the arrow points in the direction of the vector. The direction may be specified as at angle θ relative to the 0° reference.

θ

When adding only two concurrent force vectors, the resultant force may be determined by the (graphical) “parallelogram method” or by the (analytical) method of G

G

components. For two concurrent forces ( F1 and F2 ) acting on the center ring (considered as a G

G G

particle) on the force table, the resultant force, FR = F1 + F2 . To add these force vectors graphically, a parallelogram (drawn to scale) is constructed with G

G

the forces ( F1 and F2 ) as the adjacent sides with a common point (origin) where the “tails” of the vectors meet. The arrow diagonal of the parallelogram is the resultant force and its G

magnitude (length) and direction can be measured directly (as the angleφ between F1 , and the G

resultant force, FR ) from the vector diagram with a ruler and a protractor. The magnitude and direction of the resultant force may also be...

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