Force as a Vector

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TABLE OF CONTENTS

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Forces have both magnitude and direction and are therefore considered Vectors. The magnitude of a force qualifies how much force is being exerted on an object.

How force behaves

Force is exerted on objects when they interact with each other. The force ceases to exist when the interaction stops. The direction of the object's movement is also the direction in which the force is moving. Objects at rest – or in equilibrium – have opposing forces keeping them in position.

So, forces can cause motion in objects and cause objects to stay at rest. Your intuition tells you that if you want an object to move to the left, you push it to the left.

This section will introduce us to the concept of resultant force. When an object particle is subjected to a Number of forces, the resultant force is the sum of all the forces acting on the object.

Example vectors

Here are some examples of how forces can be expressed as vector quantities.

If you have two forces, $F_{1} = [\begin{matrix} 2 \\ 3 \end{matrix}] N$ and $F_{2} = [\begin{matrix} - 3 \\ 4 \end{matrix}] N$ being applied to an object, what is the resultant force?

Answer:

First, plot your forces on a graph to see their direction.

Force as a vector, Resultant force, Vaia

Figure 1. Resultant force example

If the particle at 0 is being pulled by forces 1 and 2, you can assume that the resultant force will be somewhere around the dotted line in the middle of the two forces in the diagram above. However, the question implies we should find an accurate resultant force. Moreover, other questions may not be as straightforward as this.

Resultant vector = $[\begin{matrix} 2 \\ 3 \end{matrix}] + [\begin{matrix} - 3 \\ 4 \end{matrix}]$

$= [\begin{matrix} - 1 \\ 7 \end{matrix}]$

This means that the force will end up being pulled at $[\begin{matrix} - 1 \\ 7 \end{matrix}]$ , as shown below.

Force as a vector, Resultant force, Vaia Figure 2. Resultant force

Forces can pull a particle from all angles with equal magnitude, and the resultant force is 0. This will mean the particle will be in equilibrium.

Force as a vector, Resultant force, Vaia

Figure 3. Resultant force

As demonstrated below, calculate the magnitude and direction of the resultant vector that is formed when taking the sum of the two Vectors.

Force as a vector, Resultant force, Vaia

Figure 4. Resultant force

Answer:

We break down each vector into its component form and add the components together to give us the resultant vector in component form. Then we will find the magnitude and direction of that vector.

So, we determine the x and y component of each force vector.

Let the x component of $F_{1} b e F_{1 x}$ .

And the y component of $F_{1} b e F_{1 y}$ .

$F_{1 x} = F_{1} \cos 𝛳$

$F_{1 x} = 200 N \cos (30 °)$

$F_{1 x} = 173.2 N$

Now, let's do the same with the y component.

$F_{1 y} = F_{1} \sin 𝜃$

$F_{1 y} = 200 N \sin (30 °)$

$F_{1 y} = 100 N$

Now we have the x and y component of $F_{1}$

$F_{1} = 173.2 i + 100 j$

$i$ and j are used to denote unit vectors. $i$ for vectors along the x-axis, and j for ones on the y axis.

Let's repeat the process for $F_{2}$ .

$F_{2 x} = F_{2} \cos 𝜃$

$F_{2 x} = 300 N \cos (135 °)$ [45 ° is the reference angle, but what we need is the angle relative to the positive x-axis, which is 135 °].

$F_{2 x} = - 212.1 N$

And do the same for the y component:

$F_{2 y} = F_{2} \sin 𝜃$

$F_{2 y} = 300 N \sin (135 °)$

$F_{2 y} = 212.1 N$

$F_{2} = - 212.1 i + 212.2 j$

Now that we have both forces in component form, we can add them to get the resultant force.

$F_{R} = F_{1} + F_{2}$

We will add the x components together, then the y components too.

$F_{2} = [173.2 - 212.1] i + [100 + 212.1] j$

$F_{2} = - 38.9 i + 312.1 j$

Plot this on a graph

Force as a vector, Magnitude of force, Vaia

Figure 5. Magnitude of force

Travel 38.9 units across the x-axis and 312.1 units on the y axis. That is relatively more than the length of the x-axis. The hypotenuse of the triangle formed will be the magnitude, and it has been labelled c. We use the Pythagoras theorem to find c.

It says $a^{2} + b^{2} = c^{2}$

So $\sqrt{a^{2} + b^{2}} = c$

Since c here is the same as $F_{R}$ ,

$F_{2} = \sqrt{{(- 38.9)}^{2} + {(312.1)}^{2}}$

$F_{2} = 314.5 N$

This is the magnitude of the resultant vector.

To find the direction, we will need to go back to the graph and label the angle indicated as $θ_{R}$ .

$θ_{R} = \tan^{- 1} (\frac{312.1}{38.9})$

$θ_{R}$ = 82.9 °

If you need the angle that is positive to the x-axis, you subtract 𝜃R from 180, since they are all on a straight line.

𝜃 + 82.9 = 180

𝜃 = 180 - 82.9

𝜃 = 97.1 °

Now we have the magnitude and direction of the resultant force.

Force as a Vector - Key takeaways

Force possesses both magnitude and direction.
Objects move in the direction of the net force.
Resultant force is the one force that offers the same effect to a particle as it would if it were many forces were applied.
In finding the resultant force, you add all the forces that are acting on the particle.

Frequently Asked Questions about Force as a Vector

The numerical value of the force depicts its magnitude, and the sign before it depicts its direction.

Yes

It is a free-body diagram depicting the magnitude and direction of forces acting on an object.

They can be drawn on a graph. Its magnitude is represented by the length of an arrow and its direction is represented by the direction of the arrow.

A force vector is a representation of a force that has both magnitude and direction. However, vectors do not have forces.

Flashcards in Force as a Vector12 Start learning

What is a force vector?

Force vector is a representation of a force that has magnitude and direction

Is torque a vector?

Torque is a vector quantity. Its direction depends on the direction of the force on the axis of rotation.

How do you find the direction of a force vector?

By the inverse tangent function. It is equal to the angle formed with the x-axis, or with the y-axis, depending on the application.

Can a magnitude be negative?

Magnitude is always a positive value.

What is a force vector diagram?

It is a free-body diagram depicting the magnitude and direction of forces acting on the body.

What is a resultant force vector diagram?

The diagram that represents the sum of vector forces acting on an object.

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Force as a Vector

How force behaves

Example vectors

Force as a Vector - Key takeaways

Frequently Asked Questions about Force as a Vector

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Force as a Vector

How force behaves

Example vectors

Force as a Vector - Key takeaways

Frequently Asked Questions about Force as a Vector

How do you express force as a vector quantity?

Is force a vector?

What is a force vector diagram?

How do you represent force in vector form?

What is the force of a vector?

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