Work-energy principle and conservative forces

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The discussion focuses on the work-energy principle, specifically the equation W = F * X, which is a simplified case of the integral W = ∫ F dx. It explains that when force is constant, work can be calculated as W = F (x_f - x_i), where x_f and x_i are the final and initial positions. The conversation also highlights the distinction between conservative and non-conservative forces, emphasizing that the total work accounts for all forces acting on a particle in equilibrium. Additionally, it introduces the relationship between work done by conservative forces and potential energy, expressed as Wc = -dV. Understanding these concepts is essential for grasping the equations presented.
mech-eng
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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?
 

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

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