How to Calculate Work Done by a Force: Step-by-Step Guide

📚 Understanding Work: The Physics Definition

In physics, work is defined as the energy transferred to or from an object by the application of force along with a displacement. Simply put, work is done when a force causes an object to move. It's a scalar quantity, meaning it has magnitude but no direction.

📜 A Brief History of Work in Physics

The concept of work evolved over centuries, with early ideas linked to simple machines and levers. However, a more formal definition emerged in the 19th century alongside the development of thermodynamics. Scientists like Gaspard-Gustave Coriolis and Jean-Victor Poncelet contributed significantly to defining work as a measurable quantity related to force and displacement.

🔑 Key Principles for Calculating Work

• 📏Work Formula: The basic formula for work ($W$) is given by: $W = F \cdot d \cdot cos(\theta)$, where $F$ is the magnitude of the force, $d$ is the magnitude of the displacement, and $\theta$ is the angle between the force and the displacement vectors.
• 💪Force and Displacement: Work is only done when a force causes displacement. If an object doesn't move, no work is done, even if a force is applied.
• 📐Angle Matters: The angle between the force and displacement is crucial. If the force is in the same direction as the displacement ($\theta = 0^{\circ}$), then $cos(\theta) = 1$, and the work is simply $W = F \cdot d$. If the force is perpendicular to the displacement ($\theta = 90^{\circ}$), then $cos(\theta) = 0$, and no work is done. If the force opposes the displacement, such as friction, the work is negative.
• ➕Net Work: When multiple forces act on an object, the net work done is the sum of the work done by each individual force. This is also equal to the change in kinetic energy of the object (Work-Energy Theorem).
• ⚖️Units: The SI unit of work is the joule (J), where 1 J = 1 N·m (Newton-meter).

➗ Step-by-Step Calculation Guide

• 🔍 Identify the Force: Determine the magnitude and direction of the force acting on the object.
• 📍 Determine the Displacement: Find the magnitude and direction of the object's displacement.
• 📐 Find the Angle: Measure the angle ($\theta$) between the force and displacement vectors.
• 🔢 Apply the Formula: Use the formula $W = F \cdot d \cdot cos(\theta)$ to calculate the work done. Make sure to use consistent units (Newtons for force, meters for displacement, and degrees or radians for the angle).
• ✅ Calculate: Multiply the force, displacement, and the cosine of the angle to find the work.

🌍 Real-World Examples of Work

• 🏋️ Lifting a Weight: When you lift a weight vertically, you are doing work against gravity. The force you apply is upwards, and the displacement is also upwards. If you lift a 10 N weight 2 meters, the work done is $W = 10 \cdot 2 \cdot cos(0^{\circ}) = 20 J$.
• 🚗 Pushing a Car: If you push a car horizontally, you are doing work to overcome friction. The force you apply is in the direction of motion. If you push a car with 500 N of force over 5 meters, the work done is $W = 500 \cdot 5 \cdot cos(0^{\circ}) = 2500 J$.
• 🧱 Carrying a Book Horizontally: If you carry a book horizontally while walking, the force you apply (upwards to counteract gravity) is perpendicular to the displacement (horizontal). Therefore, the work done by you on the book is zero, because $cos(90^{\circ}) = 0$.
• 🎿 Skiing Downhill: Gravity does work on a skier as they move downhill. The angle between the force of gravity and the displacement is not zero, so work is done. The amount of work depends on the skier's weight and the vertical distance they descend.

💡Tips for Calculating Work Accurately

• ✍️ Draw a Diagram: Always draw a diagram showing the force and displacement vectors to visualize the angle between them.
• 📏 Use Consistent Units: Ensure that all quantities are expressed in SI units (meters, kilograms, seconds) before performing calculations.
• ➕ Consider Multiple Forces: If more than one force acts on the object, calculate the work done by each force separately and then add them to find the net work.
• 🔎 Pay Attention to Direction: Work can be positive or negative, depending on the direction of the force relative to the displacement.

📝 Practice Quiz

Test your understanding with these practice questions:

1. A box is pushed 5 meters along a floor with a force of 25 N. If the force is applied horizontally, how much work is done?
2. A crane lifts a 100 kg object 15 meters vertically. How much work is done by the crane? (Assume $g = 9.8 m/s^2$)
3. You carry a 5 kg backpack horizontally across a 20-meter room at a constant speed. How much work do you do on the backpack?
4. A force of 50 N is applied to pull a sled across the snow. The force is applied at an angle of 30 degrees to the horizontal. If the sled is pulled 10 meters, how much work is done?
5. A constant force of 20 N is used to push a box up a ramp that is 3 meters long and inclined at 25 degrees to the horizontal. How much work is done?
6. A person pushes a lawn mower with a force of 80 N. If the handle of the lawnmower makes an angle of 40 degrees with the ground and the person pushes the mower 10 meters, how much work is done?
7. What is the work done in holding a 25 N object at rest?

✅ Conclusion

Calculating work done by a force involves understanding the force, displacement, and the angle between them. By applying the formula $W = F \cdot d \cdot cos(\theta)$ and considering real-world examples, you can master this fundamental concept in physics.