# Magnitude of the net force?

espo
the below forces act on an object; 10N north; 50 N south, and 40N west. what is themagnitude of the net force? my answer is 100 n is that correct?

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chroot
Staff Emeritus
Gold Member
Hi espo,

Think about what it means for forces to oppose each other. Let's say that you tie two pieces of rope to a bowling ball, and you and your friend tug at the ends of the rope, on opposite sides of the bowling ball. As long as you pull equally hard as your friend, the bowling ball doesn't move. If you're pulling with 40 N of force in one direction, while your friend is pulling with 40 N of force in the other direction, the net force is zero. The forces, directed in opposite directions, cancel each other out, and the ball does not move.

In the case you've given, the north and south pulls are opposite, and so cancel. They don't cancel fully, however, because they are of different strengths; the person on the south side is pulling with 40 N more force than is the person on the north.

In other words, when you have a 10 N force to the north and a 50 N force to the south, it's just the same as having a 40 N force to the south and no force at all coming from the north.

So you can combine the 10 N north and 50 N south forces into one -- by adding them -- and get one 40 N south force.

Now what is the combination -- the sum -- of a 40 N south force and a 40 N west force?

Draw a little picture of the bowling ball, with the forces being applied to it in different directions represented by arrows, like this:
Code:
               40 N

<---------*
|
|
|  40 N
|
v

The resulting force, the sum of these two forces, can be found by placing the arrows head-to-tail, like this:
Code:
               40 N

<---------*
.         |
.         |
.         |  40 N
.         |
v         v
Here, I've slid the south arrow to the left, so it's tail is on the head of the west arrow. This doesn't change the meaning of the arrow -- you can slide them around anywhere you'd like. Since they will always point in the same direction and have the same length, their meanings are not changed by sliding them around.

Now, we've got two arrows stacked head-to-tail. The first one leaves the origin, and goes a distance to the west. The second leaves the head of the first and goes some distance south. The sum of these two arrows is the arrow that connects the origin to the end of the last arrow. In other words, it's the slanted arrow shown here:
Code:
               40 N

<·········*
.        /
40 N  .      /
.    /
.  /
v/

Now, you should recognize this as a simple right triangle, with two short sides of length 40. What is the length of the hypotenuse? You can use the Pythagorean relationship:

c2 = a2 + b2

Plugging in 40 for A and 40 for B, you should find that the length of the resulting arrow has a length of about 56.5. Since in this drawing, length represents the magnitude of a force, you can conclude that the resulting force is about 56.5 newtons. Furthermore, you can conclude that the resulting force is in the south-west direction. In compass degrees, that direction is 225 degrees.

Does this make sense?

- Warren