Work done by the force

Work is a fundamental concept in physics that describes how energy is transferred when a force is applied to an object. In its simplest form, work done is the product of the force applied to an object and the distance traveled by the object in the direction of the force. In the field of mechanics, understanding work is important because it lays the foundation for more complex concepts such as energy and power.

Basic definition of work

Work is done when a force sets an object in motion. To do work on an object, some component of the force must act in the direction of motion. If force is applied but the object does not move, no work is done. Similarly, if there is motion but the force acts perpendicular to the direction of motion, no work is done either.

Work formula

The mathematical expression of the work done by a force is given by the following equation:

Work (W) = Force (F) × Distance (d) × cos(θ)

Where:

• W is the work done by the force, measured in joules (J).
• F is the magnitude of the applied force, measured in newtons (N).
• d is the distance moved by the object in the direction of the force, measured in meters (m).
• θ is the angle between the force and the direction of motion.

cos(θ) component ensures that only the portion of the force that acts in the direction of motion is considered when calculating the work done. If the force is entirely in the direction of motion, θ = 0° and cos(0°) = 1, so the full magnitude of the force contributes to the work done.

Visual example

Let us take an example to make this concept simpler.

In the above illustration, a force F is applied at an angle θ to the horizontal direction at which the object is displaced. The green arrow shows the distance d covered by the object under the influence of the force. The effective component of the force contributing to the work is along the line of motion, which is calculated using cos(θ).

Examples of work done

Example 1: Dragging a box on a flat surface

Imagine you are pulling a box on a flat surface. If you apply a horizontal force of 50 N on the box and move it 5 m along the same line, the work done will be:

W = F × d = 50 N × 5 m = 250 J

Here, the force is entirely in the direction of motion, so cos(0°) = 1.

Example 2: Lifting a box vertically

Suppose you lift a box weighing 10 kg to a height of 2 m. The force applied is equal to the weight of the box times the mass times the gravitational acceleration (about (9.8 m/s^2)). Therefore, the force is:

F = m × g = 10 kg × 9.8 m/s² = 98 N

The work done in lifting the box is:

W = F × d = 98 N × 2 m = 196 J

Here, the motion is directly in the line of force.

Example 3: Pulling a cart up an incline

Consider pulling a cart with a 30 N force applied at an angle of 30 degrees to the ground. If the cart moves 3 m along the ground, calculate the work done:

W = F × d × cos(θ) θ = 30°, cos(30°) ≈ 0.866 W = 30 N × 3 m × 0.866 ≈ 77.94 J

Here, only the horizontal component of the force acts to move the cart.

Special cases of the task

Case 1: Force perpendicular to the motion

If a force acts perpendicular to the direction of motion, no work is done. An example is carrying a book while moving horizontally. The force applied (the weight of the book) is vertical, while the motion is horizontal. Here, θ = 90°, and cos(90°) = 0, so:

W = F × d × cos(90°) = 0

Case 2: Zero displacement

If there is no movement, no work is done even if a force is applied. For example, no work is done when you push a stationary wall, as follows:

W = F × 0 = 0

Case 3: Constant velocity

When an object moves at a constant velocity and no net force acts on it, the work done is zero. This is because forces in opposite directions balance each other.

Units of work

The unit used to measure work is the joule (J). Since work is a measure of energy transfer, it is the same unit as energy. One joule is equal to the work done when a force of one newton displaces an object one meter in the direction of the force. Therefore, the relation can also be expressed as:

1 J = 1 N × 1 m

Conclusion

Understanding the concept of work done by a force is an essential step in physics that bridges the gap between force and energy. By analyzing how forces interact with motion, we gain insight into energy transfer mechanisms and how they enable us to perform various tasks in daily life. The ideas outlined here form the basis of many aspects of physics, making it important to understand these fundamental concepts well.

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