Calculating Work Done by a Force at an Inclined Angle: A Comprehensive Guide

When dealing with physical forces, understanding how to calculate the work done is crucial. This guide will walk you through solving a specific physics problem where we calculate the work done by a force that acts at an inclined angle. This type of problem is common in various fields, including engineering, physics, and mechanics. Let's dive into the details of the scenario and the mathematical approach.

The Problem Statement

Consider an object of mass 4 kg that is moved 1000 cm along a horizontal direction by a force of 20 N inclined to the horizontal at an angle of 30°. The question is: what is the work done by the force?

Understanding the Work Formula

The full formula for work is given by:

W F ? d

However, since the force is not acting along the same direction as the displacement, we need to take into account the component of the force that is in the direction of the displacement. This is given by:

F cosθ ? d

Where:

F: The magnitude of the force (N) d: The distance moved (m) θ: The angle between the force and the direction of displacement (radians or degrees)

Step-by-Step Solution

Let's break down the problem and solve it step by step:

Converting Units: Ensure that all units are consistent. The distance is given in centimeters (cm), and we need it in meters (m). 1000 cm 10 m. Component of Force: The force is inclined at 30° to the horizontal. Therefore, we need the component of the force in the horizontal direction:

F_horizontal F cosθ

Substituting Values: Substitute the values into the formula:

W (20 N) ? (10 m) ? cos(30°)

Calculation: cos(30°) √3/2 ≈ 0.866

W 20 N × 10 m × 0.866 ≈ 173.2 J

Final Answer: The work done by the force is approximately 173.2 joules (J).

Practical Application

Understanding how to solve such problems is not just theoretical but has practical applications. In engineering, for example, knowing the work done by a force helps in designing systems that require specific amounts of energy input. In physics, it is a fundamental concept that underpins more complex calculations and theories.

Conclusion

Calculating the work done by a force at an inclined angle is a critical skill in many fields of study and industry. By understanding the underlying principles and following a step-by-step approach, you can solve similar problems effectively. The key takeaway is to always consider the component of the force acting in the direction of displacement, as this is what determines the work done. Whether you are a student, engineer, or scientist, mastering this concept will be invaluable.