Vectors and Forces

Adding Forces

two pulls make one

Two forces at right angles combine by adding each direction on its own, then closing the right triangle for the resultant.

Example

Two forces at right angles combine by adding each direction on its own, then closing the right triangle to get the single resultant.

highlighted = computed this step

Two pulls at right angles

One force of 3 newtons pulls to the right and another of 4 newtons pulls straight up. What is the single force that replaces them?

Fright=3 N,Fup=4 NF_{\text{right}} = 3\ \text{N}, \quad F_{\text{up}} = 4\ \text{N}
Two forces at right anglesA rightward force arrow and an upward force arrow drawn from the same point.OF1F2

Add each direction on its own

The forces are already along the two directions, so the across total is 3 and the up total is 4.

Rx=3,Ry=4R_x = 3,\quad R_y = 4

Combine with the right triangle

The resultant closes a right triangle with legs 3 and 4: 9 plus 16 is 25, whose square root is 5 newtons.

R=32+42=25=5 NR = \sqrt{3^{2} + 4^{2}} = \sqrt{25} = \hl{5}\ \text{N}
The two forces and their resultantThe rightward and upward force arrows with the single resultant arrow along their diagonal.OF1F2R
mechanics A 3 and a 4 at right angles give a clean 5, so the resultant is exact.