7.11:
Work Done Over an Inclined Plane
Consider a person skiing down a slope. Assuming that the force due to friction on the skis is negligible over ice, a free-body diagram can be drawn.
The component of the gravitational force along the displacement moves the person forward. Forces perpendicular to the displacement have no role in the movement and, hence, do not contribute to the work done.
So, the work done is simply the parallel force component multiplied by the displacement.
Another way to solve this is by taking the force vector instead of its component.
Here, the angle between the force and the displacement is considered. Note that the final expression for work done is the same.
However, if the displacement of the skier is in the opposite direction of the force such that it starts and ends at rest, then the work done is negative.
If the angle of the inclined plane is zero, the force is perpendicular to the displacement, so the work done by the gravitational force on the person becomes zero.
7.11:
Work Done Over an Inclined Plane
The center-of-mass framework helps to easily describe the work done on rigid bodies. Since the internal forces in a rigid body do no work, they can be ignored, and the external forces can be considered in the work-energy theorem.
The work done by gravity to move a rigid body, or the work done by an opposing force to move a rigid body against gravity, can be calculated using the center-of-mass framework. It is the line integral of the force of gravity over the path, considered positive if gravity increases its kinetic energy and negative if work is done against gravity to lift it, thereby reducing its kinetic energy. In the case of horizontal motion, the work done by or against gravity is zero. That is, the change in kinetic energy of a body, even if non-zero, is purely due to other forces.
The formulation holds equally when a body moves over an inclined plane. The only difference is that only a component of gravity is along the plane, and only this component needs to be considered while calculating the work done. The gravitational force normal to the plane does no work at any point during the displacement.
Suggested Reading
- Halliday, Resnick and Walker (2014), Fundamentals of Physics, 10th Edition, Wiley and Sons Inc.: section 7.3; pages 156–157.