Understanding Negative Work While Pulling or Lifting

Work done by a force is defined as the product of the force, the displacement of the object, and the cosine of the angle between the force and the displacement. Mathematically, work ($W$) is given by:

\(W = Fd \cos\theta\)

where:

• \(F\) is the magnitude of the force.
• \(d\) is the magnitude of the displacement.
• \(\theta\) is the angle between the force vector and the displacement vector.

Work is considered negative when the force and the displacement are in opposite directions. In this case, the angle \(\theta\) between the force and displacement is 180 degrees, and \(\cos(180^\circ) = -1\). Thus, the work done is \(W = Fd(-1) = -Fd\).

Analyzing Forces During Pulling or Lifting

When an object is being pulled or lifted, its displacement typically has an upward component (when lifting) or is in the direction of the pull (which might have an upward component, like pulling up a slope).

Let's consider the forces acting on the object and the work done by each force:

• Applied force: The force applied to pull or lift the object usually acts in the direction of the displacement (or has a significant component in that direction). The angle \(\theta\) between the applied force and displacement is typically between 0 and 90 degrees. In this case, \(\cos\theta\) is positive, so the work done by the applied force is positive.
• Gravitational force: The gravitational force (weight) on the object always acts vertically downwards. When the object is being pulled or lifted upwards, the displacement is upwards. The angle between the downward gravitational force and the upward displacement is 180 degrees. Therefore, the work done by the gravitational force is negative (\(W_{\text{gravity}} = mgh \cos(180^\circ) = -mgh\), where \(m\) is mass, \(g\) is acceleration due to gravity, and \(h\) is vertical displacement). Even when pulling along a slope upwards, the vertical component of displacement is upwards, while gravity is downwards, resulting in negative work by gravity.
• Frictional force: Frictional force opposes the relative motion between surfaces in contact. If the object is being pulled horizontally along a surface, friction acts opposite to motion and does negative work. If lifting vertically without sliding along a surface, friction might be negligible or absent. The question refers to "pulling or lifting", and gravity is a force always present and opposing the upward motion inherent in lifting or upward pulling.
• Reaction force (Normal force): The normal force acts perpendicular to the surface of contact. When lifting an object straight up in the air, there is no normal force from a surface. If pulling along a surface (like a slope), the normal force is perpendicular to the surface and thus perpendicular to the displacement along the surface. The angle between the normal force and displacement is 90 degrees, so the work done by the normal force is zero (\(\cos(90^\circ) = 0\)).

Conclusion

Based on the analysis of the forces involved, the gravitational force consistently acts in the opposite direction to the upward displacement that occurs when an object is being pulled or lifted. This means the angle between the gravitational force and the displacement is 180 degrees, resulting in negative work done by gravity.

Therefore, while pulling or lifting an object, negative work is done by the gravitational force.