We
have analyzed the motion in one dimension so far,
such as the movement of a car in a straight line. We
will start to analyze the motion in two dimensions in
this chapter. There is nothing difficult in this
chapter, since the study of motion in two dimensions
is all about reducing two dimensional forces into
one.

You can combine two forces into one. Suppose Jack
pushed a box with a force of 30 N at 0 degree and
Michael pushed it with a force of 40 N at 45 degrees.
How can we find the net force acting on the box?

The first thing you have
to do is to find all forces on x direction (x axis)
only. Jack exerts 30 N and Michael exerts (cos 45 *
40) N at x direction. Therefore, the total force on x
direction would be

30 N + (cos 45 * 40) N = 58.3 N.
[E]

Then, you will have to analyze all forces on y
direction (y axis). Since Jack exerts no force and
Michael exerts (sin 45 * 40) N, the total force on y
direction would be

0 N + (sin 45 * 40) N = 28.3 N. [N]

To find the combination of Jack and Michael's
forces, we can just combine forces on x and y
directions. Therefore, using the Pythagorean Theorem,
we can calculate that

N

is the magnitude (size) of the combined forces.

QUESTION:
Find the angle of the combined force. deg

QUESTION:
If Jack exerts a force of 30 N on the box at
west and Michael 40 N at north, find the
total force exerted on the box. N

You can also break down forces. For example, Fred
pushed a box to the east and Jack pushed it to the
north. If the net force is 100 N to north east by 45
degrees,

the force applied by Fred would be

F_{Fred} = cos 45 * 100 = 70.7 N

and the force by Jack is

F_{Jack} = sin 45 * 100 = 70.7 N

QUESTION:
If the angle was 60 degrees, what is the
force applied by Fred? N

QUESTION: If
the net force was 200 N at 45 degrees, what
is the force applied by Jack? N

When you place a box on an inclined plane, the box
will slide. What is the force that makes it slide?

First, the force of gravity is acting on the box.
The force of gravity acts perpendicular to the
horizontal ground.

Also, the normal force is
acting on the box since it is on the inclined
plane. (The normal force acts on all objects on the
ground.) The normal force always acts perpendicular
to the surface, not to the horizontal. If the plane
has an incline of x degrees, then

F_{N} = Fg * cos
x

since F_{N} is leaning x degrees to the
left (Fg is the force of gravity).

There is also a force of friction between the box
and the plane. It acts parallel to the surface, not
to the horizontal.

The below drawing summarizes the forces acting on
the box:

When you combine
F_{N} and
Fg, a single force that acts parallel to the
surface will be generated. This force, called the force of parallel
(F_{//}), causes the box to move forward. F_{//}
can be calculated by Fg * sin x.

To conclude, the mixture of the force of parallel
and the force of friction determines how the box
moves. If the force of parallel is larger than the
force of friction, the box will slide. If both forces
have equal magnitude, the box will not slide. If the
force of friction is larger than the force of
parallel, the box will move upward. (Just kidding.
The force of friction can never be greater
than the force of parallel.)

QUESTION:
If the angle of the inclined plane is 30
degrees and the box has a mass of 50 kg, how
big is the F_{N} that acts on the
box? N

QUESTION: The
weight of the box is 490 N and the angle of
the plane is 30 degrees. How big is the force
that makes the box move if there is no
friction? N