1.1 Resultant Force
In our daily lives, objects are often acted upon by more than one force simultaneously, such as in a tug-of-war competition or a moving vehicle. When multiple forces act on an object, their combined effect can be represented by a single overall force known as the resultant force. Understanding resultant force allows us to determine whether an object stays stationary, moves with uniform velocity, or accelerates in a particular direction.
Learning Objectives
- Define resultant force as the vector sum of two or more forces acting on an object.
- Determine the magnitude and direction of the resultant force for parallel forces, perpendicular forces, and non-perpendicular forces using vector addition methods.
- Draw and interpret free body diagrams showing all forces acting on an isolated object.
- Relate the resultant force to an object's state of motion using Newton's Second Law of Motion (\(F = ma\)).
- Solve numerical problems involving resultant force, mass, and acceleration in various physical systems (e.g., falling objects, lifts, and pulley systems).
Determining Resultant Force for Perpendicular and Non-Perpendicular Vectors
Compare the Pythagoras rectangle, triangle of forces, and parallelogram of forces methods to find resultant force magnitude and direction.
Perpendicular Forces: Pythagoras Method
For two forces at right angles, complete a rectangle. The diagonal is the resultant force.
Non-Perpendicular Forces: Triangle & Parallelogram
Draw forces tip-to-tail for the triangle method, or tail-to-tail for the parallelogram method.
Short Explanation
What is Resultant Force?
Resultant force is the single force that represents the vector sum of two or more forces acting on an object.
Collinear Forces
When forces are in a straight line, add them algebraically by assigning a positive sign to one direction and a negative sign to the opposite direction.
Perpendicular Forces
Complete a rectangle and use Pythagoras' theorem: \(F = \sqrt{F_1^2 + F_2^2}\) and \(\tan\theta = \frac{F_y}{F_x}\).
Non-Perpendicular Forces
Construct accurate scale diagrams using the triangle of forces (tip-to-tail) or the parallelogram of forces (tail-to-tail).
| Method | Magnitude | Direction |
|---|---|---|
| Same direction (parallel) | \(F = F_1 + F_2\) | Same as both forces |
| Opposite directions (antiparallel) | Difference between larger and smaller force | Direction of larger force |
| Perpendicular | \(F = \sqrt{F_1^2 + F_2^2}\) | \(\theta = \tan^{-1}\left(\frac{F_{\text{opposite}}}{F_{\text{adjacent}}}\right)\) |
| Non-perpendicular | Measured from scale diagram | Measured with protractor |
Free Body Diagram
A free body diagram shows all the forces acting on that object only. Forces exerted by the object on its surroundings are not shown.
Newton's Second Law
According to Newton's Second Law, \(F = ma\). If \(F = 0\), the object remains at rest or travels at constant velocity. If \(F \neq 0\), the object accelerates in the direction of the resultant force.
Try to Answer First
Answer in your mind, then press "Check Answer".
What is the resultant force acting on an object if a force of \(11\text{ N}\) acts to the left, while forces of \(9\text{ N}\) and \(17\text{ N}\) act to the right?
Check Answer
Two forces of \(240\text{ N}\) horizontally to the right and \(180\text{ N}\) vertically upwards act on a ball at the same time. Calculate the magnitude of the resultant force.
Check Answer
A passenger of mass \(50\text{ kg}\) stands in an elevator that accelerates upwards at \(2.0\text{ m s}^{-2}\). Taking \(g = 9.81\text{ m s}^{-2}\), what is the normal reaction \(R\) exerted by the floor on the passenger?
Check Answer
Common Mistakes
- Forgetting that force is a vector quantity and adding magnitudes directly without considering angles or directions (e.g., adding \(6\text{ N}\) and \(8\text{ N}\) perpendicular forces to get \(14\text{ N}\) instead of \(10\text{ N}\)).
- Confusing "constant velocity" with having a non-zero forward resultant force. When velocity is constant, acceleration is zero, so the resultant force must be zero (\(F = 0\text{ N}\)).
- Drawing forces that the object exerts on other bodies in a free body diagram, rather than only the external forces acting on the isolated object.
- Omitting the tension or normal reaction when setting up \(F = ma\) equations for vertical motions in lifts or falling objects with resistance.
Concept Misunderstandings
An object moving forward always has a net force acting forward in the direction of motion.
An object moving forward at a constant speed has a resultant force of zero (\(F = 0\text{ N}\)). A forward resultant force is only required to accelerate or change velocity.
The normal reaction of an object in a lift is always equal to its true gravitational weight (\(mg\)).
The normal reaction \(R\) equals \(mg\) only when the lift is stationary or moving with uniform velocity. When accelerating upwards, \(R = mg + ma\) (apparent weight is greater); when accelerating downwards, \(R = mg - ma\) (apparent weight is smaller).
Summary
- Resultant force is the vector sum of two or more forces acting on an object.
- Collinear forces are combined by algebraic addition with directional sign conventions.
- Perpendicular forces are solved using Pythagoras' Theorem (\(F = \sqrt{F_1^2 + F_2^2}\)) and \(\tan\theta = \frac{F_y}{F_x}\).
- Non-perpendicular forces are determined using the triangle of forces or parallelogram of forces method.
- A free body diagram isolates an object to show all external forces acting directly on it.
- \(F = ma\): If \(F = 0\), velocity is constant or zero (\(a = 0\)); if \(F \neq 0\), the object accelerates in the direction of the resultant force.
Short Activity
Objective: Identify resultant force, free body diagrams, and Newton's Second Law.
A. Objective Quiz
1 Which of the following conditions is true for an object moving with a uniform velocity?
2 Two perpendicular forces of \(6\text{ N}\) and \(8\text{ N}\) act on an object. What is the magnitude of the resultant force?
B. Fill in the Blanks
3 The single force that represents the vector sum of two or more forces acting on an object is called the .
4 In a vector force table experiment, as the angle between two applied forces increases, the magnitude of their resultant force .
C. Matching / Drag and Drop
Match each state of motion of an object to its corresponding resultant force condition. If using a phone, tap the answer first, then tap the matching box.
