How Do You Minimize Work on an Inclined Plane?

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Minimizing work on an inclined plane involves understanding the relationship between work, force, and displacement. The equation for work is defined as Work = Force x Displacement. To achieve minimum work, the net force acting on the object must also be minimized. By analyzing the Free Body Diagram, the equation 0 = ma = F - mg sin α - μ mg cos α can be derived. Differentiating the force with respect to the angle α and setting it to zero provides the necessary conditions for minimizing work.
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Work is Force X Displacement. Now Work is minimum when the net force is minimum
By drawing the Free Body Diagram for the block, you would arrive at
0=ma=F - mg \sin \alpha - \mu mg \cos \alpha
Differentiating F with respect to alpha and equating to zero you would get the required result
 

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