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"Scalars and Vectors"

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Lucas grabs the handle of a cart and pulls it up a slope. The cart is heavy and Lucas has a tough time moving forward. So, he calls his friend John for help. Now, the cart can move with ease and is pulled to the top of the slope without much effort.

Lucas and John acted on the cart each with their forces. The two forces added up to produce a bigger force with a greater total effect on the cart than if Lucas were towing alone.

Let’s look at another example where two forces act on one body in different directions.

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We see that the forces acting together on the body add up and cause a joint effect on it. Below, we look at what is the resultant of adding forces and how we add forces.

When several forces act on a body at the same time, we can add them up. As a result of the addition, we get a force, which we call the sum of forces or in one word: **resultant**. It replaces all the forces that previously acted on the body.

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The resultant is therefore a force. Like any force, the resultant can be represented by a directed line. We will give it:

*direction*using an arrow and*magnitude*using a scale.

The resultant is the sum of all the forces acting on a body. It is a single force with which we replace all other forces. Its effect on the body movement is the same as the effect of all forces together.

Let's look at how we add forces. It is different if the forces act in the same directions or in different directions. Therefore, we will consider the following examples when determining the resultant:

parallel forces acting on a body in the same direction,

parallel forces acting on a body in the opposite direction,

forces act on a body in different directions.

Parallel forces acting on a body can have the same or opposite direction:

In all cases of parallel forces, the resultant can be obtained by:

graphical addition of forces and

computational addition of forces.

We will discuss both methods in more detail below.

The sum or resultant of parallel forces can be obtained in two ways:

with graphical addition of forces,

with computational addition of forces.

If several forces act on a body in the same direction, then their combined effect increases. The effect is as if one force that has the magnitude of all the forces together was acting on the body.

Let’s look at the graphical and computational processes of the addition of forces. We will look at both procedures using the case of two forces. If there are several forces, we add up all the forces according to the same procedures.

**Calculating the resultant of forces acting in the same direction by graphical method**

We add forces graphically by taking the following steps:

We first choose a scale in order to know the length of the line that will represent each force.

Next is to draw the forces at the exact points where they act on the body.

Then the forces are moved in their parallel positions from the body.

We then connect the forces in such a way that the beginning of one touches the end of the next and so on in that order.

- Now we obtain the:
direction of the resultant by drawing a straight line in the direction of all the forces and equal in length to the total length of the forces.

The magnitude of the resultant is represented by the length of the line and obtained by applying the scale.

We can arbitrarily choose any force to be the first or second or third and so on.

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**Calculating the resultant of forces acting in the same direction by computational method**

The forces act in parallel and in the same direction, so we can also get the result by simply adding the magnitudes of the forces. The resultant points in the same direction as both forces.

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When forces act on a body in opposite directions, their effects also contradict each other. It is as if one force was acting on the body with a magnitude equal to that of the difference between the magnitudes of the two forces. We say the forces are subtracted.

Let’s look at the graphical and computational methods of adding forces acting in the opposite direction.

**Calculating the resultant of forces acting in the opposite direction by graphical method**

We add the forces graphically by taking the following steps:

We first choose a scale in order to know the length of the line that will represent each force.

Next is to draw the forces at the exact points where they act on the body.

Then the forces are moved in their parallel positions from the body.

We then connect the forces in such a way that the beginning of one touches the end of the other.

- Now we obtain the:
direction of the resultant by drawing a straight line in the direction of the greater force and equal in length to the difference in the lengths of the forces.

The magnitude of the resultant is represented by the length of the line and obtained by applying the scale.

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**Calculating the resultant of forces acting in opposite direction by computational method**

The forces act in parallel and in the opposite direction. Therefore, the resultant can also be obtained by subtracting the forces numerically. The resultant points in the **direction of the greater force**.

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Two forces are equal and opposite if they:

have equal magnitudes and

are oppositely directed.

In this case, they cancel out each other (deducted). Their resultant is zero, so the body does not move.

When calculating the sum of two opposing forces, we assign them a positive and a negative sign, for example:

the force directed to the right has a positive sign and

the force directed to the left has a negative sign.

The forces in the figure above are equally large but directed in the opposite direction. Their sum or resultant is therefore equal to zero:

The effect of both forces on the motion of the body is the same as if no force were acting on it.

If the forces are equally large and directed in the opposite direction, their sum or resultant is equal to zero.

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Non-parallel forces act on a body in different directions. The grip of non-parallel forces is not necessarily at the same point. By summing such forces we also get a resultant that replaces all the forces.

The size of the resultant is not equal to the sum of the magnitudes of all the forces. We need to keep in mind that the forces are acting in different directions. Their magnitudes therefore do not add up as numbers.

Non-parallel forces are summed geometrically. There are two possible methods (rules) for addition:

parallelogram rule or

polygon of forces

In both methods, we use parallel placements of forces and obtain the same resultant.

When summing non-parallel forces, we use a parallelogram rule or form a polygon of forces, both with parallel movement of forces.

We mostly use this rule when summing two non-parallel forces.

The procedure for adding forces using the parallelogram rule is as follows:

We arrange the two forces exactly the way they act on the body and in their directions to make two of the sides of the parallelogram.

We produce the other two sides by drawing lines parallel to and in the same directions as the first two forces.

The resultant is the diagonal of the parallelogram with:

its beginning at the same point as the point at which the two forces act on the body

and an arrow at the intersection of the parallel lines.

Let's look at the procedure in a concrete case.

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This procedure is used when adding two or more non-parallel forces at once.

If we add only two forces, we call this procedure the *triangle rule*, because the forces form a triangle. Typically, however, this procedure is used when adding several non-parallel forces. Then we get a polygon of forces.

The process of adding forces using this method is such that we move the forces in their directions and load them on top of each other. This means we arrange the forces end-to-end according to their directions.

The resultant is a directed line that connects the beginning of the first force and the end of the last force in the arrangement.

We will look at the procedure in a concrete case.

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material editor: Habeeb Adenle