- Scalar quantities and
- Vector quantities
Vector quantities need more description than magnitude alone. Forces are vector quantities. These quantities possess magnitude and direction.These are known as vector quantities and include such things as force, displacement, velocity and acceleration. Vector quantities possess both magnitude and direction and can be represented by a line drawn to scale in a certain direction with an arrowhead indicating its sense. This line is called a vector (see Figure 1.2).
Scalars are added algebraically (for example, the distances of 5 m + 7 m are simply
added to give 12 metres); however, vectors must be added geometrically because their
line of action and direction are just as important as their magnitude. When vectors
are added, the solution is known as the resultant. This is the one vector which could
replace all the others without changing the overall effect. If the vector diagram ABC is drawn to a suitable scale, the resultant vector AC will be to the same scale, and the magnitude of this resultant vector can easily be obtained by measuring the length of the vector (line) AC. This method works equally well for other vector quantities. For example, a force has magnitude, line of action, direction, and point of application and all of these factors can be represented by a single line as shown in the following diagram
An interaction that causes an acceleration of a body is called a A - The vector sum of all the external forces that act on a body
**must be zero** - The vector sum of all the external moments that act on a body
**must be zero**
These vector equations, are each equivalent to three independent scalar equations, one for each direction of the coordinate axes: - Sum of all vertical forces must be zero
- Sum of all horizontal forces must be zero
**Sum of all rotational (moments) forces must be zero**
The The gravitational acceleration on Earth is ~9.81 m/s². However, in our
subject we round this figure up to 10 m/s². This fits well the
decimal system and makes conversion from kilogram All this can be summerised in a simple vector equation, which is called Newton's second law of motion:
The weight of a body on earth is the W = m x gg = acceleration due to gravity. The gravitational acceleration is
9.8 m/s². In our subject we round this up to 10 m/s². This
figure fits the decimal system and make the convertion from a mass unit
to a weight unit much easier.
ForcesForces are represented by a line drawn to a force-scale in a certain direction with an arrowhead indicating it sense as shown below. A force is completely defined by its:
Concurrent coplanar force systemsCoplanar means in the same plane like the force shown above drawn on a sheet of flat paper. Forces that lie in the same plane are two-dimensional with a x-direction and a y-directions. A concurrent coplanar force system is a system of two or more forces whose lines of action all intersect at a common point. However, all of the individual vectors might not actually be in contact with the common point. (as can be seen in this diagram)
The Cartesian plane is used for the graphical “resolution” of forces. As all forces are vectors their magnitude and direction is easily recognised if they are drawn to scale in the Cartesian plane as shown below.
The parallelogram of forces method is a graphical methods to find the resultant of a coplanar force system. Two or more concurrent forces can be replaced by a single resultant force that is statically equivalent to these forces.
The illustration above shows two force vectors. In order to resolve these forces graphically, one must first extend the lines of action of two concurrent forces until they intersect.
The
5.8 kN acting at 30° to the horizontal to the right and 3.5 kN acting at 20° to the vertical to the left
(a) graphical
First select an appropriate force scale (e.g. 10 mm = 1 kN). Next step would be to draw the forces at the selected scale on graph paper. Remember to add them tail-to-head. Connect the tail end of the first drawn force with the head of the last drawn force an measure the distance. The distance in millimeter divided by 10 gives you the magnitude of the resultant in kN.
In general we use the sin and cos function to calculate the components of the forces and add them together. Whether you have to add or subtract the components depents on the direction (see geometrical coordinates). We can then use Pythagors theorem to calculate the resultant.
Having calculated the x and y component we can now use Pythagoras theorem to calculate the resultant. In the above table the reference angle must always refer to the horizontal [sinA=cos(90-A)] To find the angle of the resultant in reference to the horizontal we use the tan-function.-
By using the invert funtion of your calculator you find the angle of 58.28°
to the horizontal.
consider the above forces F
The calculation refers to the cos of the angle between the 5.8 kN force and the resultant R = 7.276 kN. The inverse of cos = 0.8809 is 28.25°. The angle of the 5.8 kN to the horizontal (30°) must be added and the is 58.25°, which worked out the same as the other figures.
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