Calculating Acceleration with Perpendicular Forces: A Step-by-Step Guide

Understanding how to calculate acceleration when two perpendicular forces act on an object is a fundamental concept in physics. This tutorial will walk you through a comprehensive example, detailing the application of the Pythagorean Theorem, trigonometry, and Newton's Second Law.

Understanding the Problem

In this scenario, we have two forces, 30N and 40N, acting on a body of mass 25kg, with these forces being perpendicular to each other. Our goal is to determine both the magnitude and direction of the acceleration of the body. This problem involves several key physics principles and mathematical techniques.

Calculating the Resultant Force

When dealing with perpendicular forces, we can use the Pythagorean Theorem to find the magnitude of the resultant force. According to the Pythagorean Theorem, for a right-angled triangle, the square of the hypotenuse (the resultant force) is equal to the sum of the squares of the other two sides.

Magnitude of the Resultant Force

Resultant Force (FR) √(F12 F22)
FR √(302 402)
FR √(900 1600)
FR √2500
FR 50N

The magnitude of the resultant force is 50N. This result shows us the total force acting on the body, taking into account both individual forces.

Direction of the Resultant Force

Next, we need to determine the direction of the resultant force. This can be done using trigonometry, specifically the tangent function, which helps us find the angle θ between the resultant force and one of the individual forces.

To find the direction, we use the following formula:

tan(θ) opposite/adjacent F2/F1
tan(θ) 40/30
θ arctan(40/30)
θ ≈ 53.13°

The direction of the resultant force is approximately 53.13° from the force of 30N. This is important because it tells us the angle at which the combined force is acting.

Determining the Acceleration

Once we have the magnitude and direction of the resultant force, we can use Newton's Second Law of Motion to determine the acceleration of the body. Newton's Second Law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration: F ma.

Given the force (FR) and the mass (m), we can solve for acceleration (a):

a FR/m
a 50N / 25kg
a 2m/s2

Conclusion

In conclusion, the magnitude of the resultant force acting on the body is 50N, and the direction of this force is approximately 53.13° from the force of 30N. Using this, we determined the acceleration of the body to be 2 m/s2.

Understanding and solving such problems not only helps in grasping the fundamental principles of physics but also enhances problem-solving skills in various real-world applications, such as in engineering and vehicle design.