Resultant Force

Resultant force examples

The resultant force on an object is the vector sum of all of the forces acting on the object. This comes from the resultant force definition and can be illustrated with the following examples.

Resultant force equation

We can write a general equation for the resultant force$F_{res}$on an object when multiple forces act on it in a straight line. Suppose we have two forces,$F_1$and$F_2$acting on an object in the same direction; the resultant force equation can be written as the sum of each of the forces as follows.

$F_{res}=F_1+F_2$

If two forces,$F_3$and$F_4$, act in a straight line but in opposite directions on an object, the resultant force equation is written as the difference between the two objects as below.

$F_{res}=F_3-F_4$

Let's look at an example of multiple forces on an object in different directions. The corresponding resultant force diagram can be seen in the figure below.

The resultant force on an object is the vector sum of all the forces acting on that object as shown by this diagram. - StudySmarter Originals

$F_1$and$F_2$act in the same direction and$F_3$acts in the opposite direction to$F_1$and$F_2$. The resultant force equation is, therefore:

$F_{res}=F_1+F_2-F_3$

The magnitude of resultant force

Recall that force is a vector quantity, so we use minus to indicate its direction, e.g. if we choose the right direction to be positive, any force to the left has a minus sign in front of its value. The magnitude of a resultant force $F_{res}$, is a scalar quantity, so in the case of multiple forces acting on an object, the magnitude of the resultant force is just the size of the force without its direction. That is, we drop the minus sign in front of the number (if there is one). An example of this is given below.

Resolving forces: Resultant force diagrams

From the example above, we can see that there is no resultant force when the forces on an object are balanced. Also, the direction of the resultant force on an object gives us the direction in which the object will move. All of the forces mentioned above are applied parallel to the floor or surface along which the object is able to move. What would happen if the force on an object were applied at some angle to the surface?

Imagine that a box is being pushed across a floor by a force$F$of magnitude$20\text{N}$, but the force is applied downward and to the right at an angle of 35° to the horizontal. We have to resolve the force into components, since the box moves horizontally along the surface.

A force F is applied to a box at an angle of 35° to the horizontal surface. The box moves horizontally along the surface. - StudySmarter Originals

We can draw a resultant force diagram to resolve$F$into a horizontal and vertical component as follows (not drawn to scale).

A 20 N force is resolved into its horizontal (16 N) and vertical (11 N) components. - StudySmarter Originals

If the arrows in the diagram are drawn accurately and to scale, we can measure the vertical component of the force (by measuring the length of the vertical arrow) to be$11\text{N}$and the horizontal component (by measuring the length of the horizontal arrow) to be$16\text{N}$. We must make sure to draw them to scale. Our arrows should form a right-angle triangle, when the arrows representing the horizontal and vertical components of the resultant force are placed head to tail.

The vertical component of the force balances the normal contact force between the floor and the box. The horizontal component is responsible for moving the box horizontally across the floor. If we can resolve forces to find the components responsible for causing objects to move, we can find the resultant force for any number of forces acting on an object.