Work-energy principle and conservative forces

[mech-eng]

Hi, all there are equation in the pic but I can't understand them. I know work-energy principle which
is W= F * X (work equals force times way) but I think they are special forms. What concepts
and topics should I study to understand them?

 

[Pythagorean]

W = F*x is a special case of the integral:

$$W = \int F dx$$

if F is not dependent on x, then we can write:

$$W = F \int_{x_i}^{x_f} dx$$

and that's just

$$W = F (x_f - x_i)$$

The x you use is the distance moved, which is just the difference between the final and initial position, as I have written.

The equation you show accounts for all forces on a particle and breaks them into conservative and non-conservative forces and assumes the particle is in equilibrium, setting that sum of forces to zero.

 

[dauto]

The text also makes use of the definition of potential energy difference dV in terms of the work done by a conservative force W_c= - dV