The coefficient of static friction between a block and a horizontal floor is 0.40, while the coefficient of kinetic friction is 0.15. if a horizontal force pulling the block is slowly increased until it is barely enough to make the block start moving, what is the acceleration of the block as it starts to move?

aceleration as start the move is  2.45  m/s²

Explanation:

We use Newton's second law on each axis. In the x-axis we have two forces the driving force and the friction force in the opposite direction, in the axis and we also have two forces the normal one up and the weight down.

 To find the driving force we use the equilibrium condition just before the movement begins, in this case the friction coefficient is static.

X-axis        F-fr = 0   ⇒     F = fr

Y-axis       N-w = 0   ⇒   n = w = mg

The expression for the friction force is fr = μ N

      Fr = μs m g

      F = μs m g

μs  is the coefficient of static friction

Now it starts moving and some of the links are broken, which is reflected in the fact that the coefficient of kinetic friction is lower, if we keep in value of the driving force, Newton's second law remains

       F -fr = ma

       N-w = 0     ⇒     N = w = mg

       fr = μk N = μk m g

μk is the coefficient of kinetic friction

       a = (F -fr) / m

       a = (μs m g - μk m g) / m

       a = (μs - μk) g

       a = (0.4 - 0.15) 9.8

       a = 2.45  m/s²

  This is the acceleration due to the decrease in the friction coefficient

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