Consider a particle that describes a curvilinear trajectory (C) under the action of a force F, as shown in the figure below. The elementary displacement vector dr is tangent to the trajectory at each point. θ is the angle between F and dr.
The work done by a force F on the particle when it moves between the points A and B of its trajectory is given by:
As seen from the previous expression, the work is a scalar quantity and is measured in Joules (J) in the International System.
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In general, the work done by a force depends on the trajectory described by the particle, since the angle between the force and elementary displacement vectors depends on it; however, as we will see later, when the force is conservative, its work depends only on the coordinates of the initial and final points of its trajectory.
When more than one force acts on a particle, the total work is the sum of the works of the forces, each one with its sign.
Kinetic energy
Next, we will develop the work expression to see what is its effect on a particle’s motion.
First, we develop the dot product that is inside the integral:
As you can see in the following figure, Fcosθ is the projection of the force on the axis tangent to the trajectory:
And using Newton’s second law we get:
Where we have replaced the magnitude of the tangential acceleration .
Since magnitude of the displacement vector is approximately equal to the arc traveled by the particle:
The kinetic energy of the particle is:
Therefore, the work of the force is equal to the change in the kinetic energy of the particle.
When several forces act on the same particle:
Potential energy. Conservative forces
A conservative force can be derived from a scalar function called the potential energy using the gradient operator:
In addition, the work of a conservative force depends only on the initial and final coordinates of the particle’s position and is equal to minus the change of the potential energy:
Therefore, if the force acting on a particle is conservative we can equate the two ways of calculating its work:
And after grouping the terms we get the principle of conservation of energy:
As examples of potential energy we can mention the one associated with the weight ( gravitational potential energy):
and the one associated with the spring force (elastic potential energy):
Both kinetic and potential energy are measured in Joules (J) in the International System.
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