Understanding Negative Work in Physics: When Force and Displacement Are More Than 90 Degrees
Understanding the concept of work in physics particularly when the angle between force and displacement is more than 90 degrees is crucial. The work done, mathematically defined, can be either positive or negative, depending on the angle. This article delves into the specifics of negative work, providing a clear explanation and practical examples.
Understanding Work in Physics
Work is defined in physics as the product of force, displacement, and the cosine of the angle between the two vectors:
Mathematical Definition of Work
Mathematically, work (W) is given by the dot product formula:
[W vec{F} cdot vec{d} costheta]
Here:
(W) Work done (vec{F}) Magnitude of the force (vec{d}) Magnitude of the displacement (theta) Angle between the force and displacement vectors.When the Angle is More Than 90 Degrees
When the angle (theta) between the force and displacement vectors is more than 90 degrees (i.e., (theta > 90^circ)), the cosine of the angle becomes negative. This results in negative work, indicating that the force is contributing in the opposite direction of the displacement.
Explanation of Negative Work
For (theta > 90^circ):
The cosine of the angle (theta) is negative. Since the magnitudes of the force ( vec{F} ) and displacement ( vec{d} ) are always positive, the negative value of (costheta) will make the overall product ( vec{F} cdot vec{d} costheta ) negative. Therefore, the work (W) done is negative.Example 1: Object Pushed Against a Wall
Consider the scenario where you push an object horizontally to the right but the object does not move because it is blocked by a wall. The force vector ( vec{F} ) and the displacement vector ( vec{d} ) are as follows:
( vec{F} 10 , text{N} ) (horizontal to the right) ( vec{d} 0 , text{m} ) (displacement is zero because the object does not move)Since there is no displacement ((vec{d} 0 , text{m})), the work done is:
[W 10 , text{N} cdot 0 , text{m} cdot cos(120^circ) 10 , text{N} cdot 0 , text{m} cdot (-0.5) 0 , text{J}]
However, for a scenario where the object is pushed at an angle of 120 degrees from the horizontal direction of the displacement, the angle (theta 120^circ), and the displacement is (5 , text{m}), the work done can be calculated as:
[W 10 , text{N} cdot 5 , text{m} cdot cos(120^circ) 10 , text{N} cdot 5 , text{m} cdot (-0.5) -25 , text{J}]
This negative value indicates that the force is doing negative work, meaning that the object loses energy as the force works against the displacement.
Example 2: Force Applied at 135 Degrees
Consider another scenario where a force ( vec{F} ) is applied at an angle of 135 degrees from the horizontal direction. Here, (theta 135^circ), and let's assume the force is (10 , text{N}) and the displacement is (5 , text{m}).
The cosine of 135 degrees is (-frac{1}{sqrt{2}}), thus the work done is:
[W 10 , text{N} cdot 5 , text{m} cdot left(-frac{1}{sqrt{2}}right) 10 , text{N} cdot 5 , text{m} cdot left(-0.707right) -35.35 , text{J}]
Again, the negative value indicates that the work done is negative, meaning the force is doing negative work on the system.
Conclusion: In the context of physics, if the angle between the force and displacement is more than 90 degrees, the work done is negative. This means the force is contributing in the opposite direction of the displacement, resulting in a loss of energy in the system.
Additional Tips and Insights
Understanding the relationship between the angle and the work done is essential in various applications, including mechanical engineering, physics, and physics-based simulations. Negative work can represent various physical phenomena, such as friction, air resistance, or energy dissipation.
It is also worth noting that the concept of negative work is not limited to mechanical systems but can be extended to other fields such as electrical and thermal systems where work can be stored or dissipated in a similar manner.
Keywords: negative work, angle between force and displacement, work in physics