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Chapter 5 Motion in Two Dimensions

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.


1. Simple Breakdown of Forces
2. Two Dimensional Forces into One
3. One Dimensional Forces into Two
4. Forces involving Gravity
5. Forces in Three Directions
6. Chapter 5 Quiz

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Section 1: Simple Breakdown of Forces

You can break down forces into several components easily. For example, the force F1 can be broken into two forces: Fx and Fy.

Illustrating breakdown of a force

The following formulas are true:



Fx = cos A * F1
Fy = sin A * F1

The idea of "breaking down forces" is very important in this chapter.



Section 2. 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?

Michael and Jack pushing a 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


is the magnitude (size) of the combined forces.

QUESTION: Find the angle of the combined force.

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.



Section 3: One Dimensional Forces into Two

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,

Jack and Fred pushing a box

the force applied by Fred would be

FFred = cos 45 * 100 = 70.7 N

and the force by Jack is

FJack = sin 45 * 100 = 70.7 N

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

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



Section 4. Forces involving Gravity

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

A box on an inclined plane

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



FN = Fg * cos x

since FN 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:

Forces acting on a box on inclided plane



When you combine FN 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.

Calculating the force of parallel

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 FN that acts on the box?

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?



Section 5. Forces in Three Directions

If you see a mixture of three or more forces like below,

Forces in three directions

All you have to do is to calculate forces on x direction, on y direction, and add these two forces into one to get the total net force.



Section 6: Chapter 5 Quiz

Try the Chapter 5 Quiz and see how well you can do!

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