Answer:
To determine the resultant and direction of the forces below, we need to use vector addition. Vector addition is the process of finding the sum of two or more vectors by adding their components. A vector has both magnitude and direction, and can be represented by an arrow. The length of the arrow is proportional to the magnitude of the vector, and the direction of the arrow indicates the direction of the vector.
To add vectors, we can use either the parallelogram method or the component method. The parallelogram method involves drawing the vectors as adjacent sides of a parallelogram, and then drawing the diagonal from the common point as the resultant vector. The component method involves breaking down each vector into its horizontal and vertical components, and then adding them separately to find the resultant vector.
In this case, we will use the component method, since it is easier to calculate. We will assume that the positive x-axis is to the right and the positive y-axis is upward. We will also use degrees as the unit of angle measurement.
The forces given are:
3 N at 60°
4 N at 0°
7 N at 20°
9 N at 50°
To find the components of each force, we use trigonometry. The horizontal component of a force is given by F cos θ, where F is the magnitude of the force and θ is the angle it makes with the x-axis. The vertical component of a force is given by F sin θ, where F is the magnitude of the force and θ is the angle it makes with the x-axis.
The components of each force are:
3 N at 60°: horizontal component = 3 cos 60° = 1.5 N, vertical component = 3 sin 60° = 2.6 N
4 N at 0°: horizontal component = 4 cos 0° = 4 N, vertical component = 4 sin 0° = 0 N
7 N at 20°: horizontal component = 7 cos 20° = 6.5 N, vertical component = 7 sin 20° = 2.4 N
9 N at 50°: horizontal component = 9 cos 50° = 5.8 N, vertical component = 9 sin 50° = 6.9 N
To find the resultant force, we add up all the horizontal components and all the vertical components separately. The resultant force is then given by:
Horizontal component = (1.5 + 4 + 6.5 + 5.8) N = 17.8 N
Vertical component = (2.6 + 0 + 2.4 + 6.9) N = 11.9 N
Resultant force = sqrt((17.8)^2 + (11.9)^2) N ≈ 21.5 N
To find the direction of the resultant force, we use inverse trigonometry. The direction of the resultant force is given by tan^-1 (vertical component / horizontal component), where tan^-1 is the inverse tangent function that gives the angle whose tangent is equal to a given value.
The direction of the resultant force is then given by:
Direction = tan^-1 (11.9 / 17.8) ≈ 34°
Therefore, the resultant force is about 21.5 N at an angle of about 34° from the positive x-axis.
You can watch some videos on how to calculate resultant forces and directions using vector addition here:
A-Level Maths: R1-12 Forces: Finding the Magnitude and Direction of a Resultant Force
GCSE Physics - Resultant Forces & Free Body Diagrams #42
Resultant Forces | Force & Motion | Physics | FuseSchool
You can also read some web pages on how to find resultant forces and directions using vector addition here:
What is the Resultant Force and How to Find it (with Examples)
Vector word problem: resultant force (video) | Khan Academy
How do you find the direction of the resultant force?
Calculations involving forces - Force and Newton’s laws - BBC
How to calculate the magnitude and direction of resultant force?
I hope this helps you understand how to find resultant forces and directions using vector addition. If you have any other questions, please feel free to ask me.