Understanding Particle Displacement: A Case Study
When dealing with mechanical movements, understanding the concept of displacement is crucial. This article delves into the specific scenario where a particle undergoes two separate displacements, and we calculate the resultant displacement using vector addition. We will also discuss the significance of this computation in the field of physics.
Introduction to Particle Displacement
A particle is an idealized object in physics that has mass but no size or shape. When we talk about the displacement of a particle, we are referring to the change in its position from an initial point to a final point. Displacements can be represented using vectors, which have both magnitude and direction.
Given Displacements
Let's consider a specific scenario where a particle undergoes the following displacements:
A displacement of 5 cm in the east direction (positive x-axis). A displacement of 4 cm at an angle of 60 degrees north of east (positive x-axis and positive y-axis).The goal is to find the magnitude and direction of the resultant displacement by adding these vector displacements.
Calculating the Resultant Displacement
The position vector of the particle can be expressed as: R 5i 4cos60i 4sin60j
First, let's simplify the components:
i component: 5 4cos60 5 4 * 0.5 7 j component: 4sin60 4 * 0.866 3.464Therefore, the position vector becomes:
R 7i 3.464jTo calculate the magnitude of the resultant displacement, we use the Pythagorean theorem:
| |R| √(72 3.4642) √(49 11.99) √60.99 ≈ 7.81 cmThe magnitude of the resultant displacement is approximately 7.81 cm.
Determining the Direction of the Resultant Displacement
The direction of the resultant displacement can be determined using the arctangent function to find the angle θ between the resultant vector and the x-axis (east direction).
θ tan-1(3.464/7) ≈ 26.329°This angle is measured north of east. However, it is more conventional to express this direction as east of north. Therefore, the direction is approximately 63.671° east of north.
Conclusion and Significance
This case study demonstrates the practical application of vector addition in physics. Understanding resultant displacement is fundamental for analyzing more complex scenarios in mechanics and the behavior of particles. The skills used here are applicable in various fields, including engineering, robotics, and astronomy, where precise measurement and calculation of movements are vital.