A person pulls on a block with a force , at an angle with respect to the
horizontal. The coefficient of friction between the block and the ground is .
For what is the required to make the block slip a minimum?
Problem Source : Introduction to Classical Mechanics
The free body diagram can be drawn as below:
What does slip a minimum means? It yields that the acceleration of the block . Therefore, the friction force is a kinetic friction but the net force is . With this in mind, we can now construct our equation:
Taking the derivatives of respect to ,
Hence, is the required to make the block slip a minimum.