Physics question on scalars and vectors?

Question

I encountered this question and can't remember how I solved it.

"A force of 3.0N and a force of 4.0N act on an object at the same time. Which of the forces cannot be the resultant force on the object?"

There are 4 options (0.5N, 3.5N, 4.0N, 5.0N)

I chose 4.0N but the correct answer is 0.5N.

Can someone explain why my answer is wrong and why 0.5N is correct? 10 points will go to the best answer. Thanks in advance.

Answer 1

The smallest resultant would be 1.  If you draw a vector 4 units long and make a 3 unit circle at the end of it, you have the possible resultants (from the base of the 4 to the tip of the 3), since you haven't been give direction.  If the 3 directly opposes the 4, you will get the smallest possible resultant   = 1.

Answer 2

When you add two vectors, then the minimum value of the magnitude of the resultant vector = difference in magnitudes of the vectors

and the maximum value of the magnitude of the resultant vector = sum of the magnitudes of the vectors.

The vectors have magnitudes 3.0N and 4.0N

Therefore, the minimum magnitude of the resultant = 4.0N - 3.0N = 1.0N

The maximum magnitude of the resultant = 4.0N + 3.0N = 7.0N

This means that the resultant must be between 1.0N and 7.0N

It cannot be less than 1.0N and it cannot be greater than 7.0N

0.5N is not possible because it is less than 1.0N.

Ans: 0.5N

Answer 3

0.5. because if they were directly opposite eachother the resultant force would be 1.0N in any other position the forces would be larger!

Therefore it could not be 0.5N but could be any of the others.

Answer 4

Since the forces can be acting in any direction on the object, the magnitude of the resultant force can have a wide range of values.

An easy way to see this is to draw a circle of radius 4. Now any line drawn from the centre of this circle to its circumference will represent a force of 4 N. Now imagine at any point on this circle's circumference, you draw another circle of radius 3. Any line from the centre of the first circle to the circumference of the second represents your resultant force.

If you have done the above it is clear there is a limit to how small this resultant force can be. In fact its smallest possible value is 4-3 = 1N. Therefore 0.5 cannot be the resultant force. Conversely, the fact that your second circle intersects with the first circle shows that your resultant force can end up with a value of 4N. To find out if 5N or 3.5N is possible, you can draw a circle of radius 3.5 or 5 with the same centre as your first 4N circle. If your second 3N circle drawn on the circumference can intersect these 3.5 and 5 circles, then they are valid possible resultant forces.

Of course you can also do all of the above mathematically with vector algebra.