Determine the maximum value of the coefficient of friction:

A force of 200 N inclined at 60^o to the horizontal is applied to the block A weighing 400 N. find out whether block A moves if the coefficient of friction is 0.5. If not, then determine the maximum value of the coefficient of friction while it is just on the point of moving.

 Solution

There is a chance of movement of block A towards right. Thus, the direction of frictional force FA will be towards left.

N _A  = W - P sin 60^o

= 400 - 200 sin 60^o = 226.795 N

 F_A ( max )  = μ N A

= 0.5 × 226.795

= 113.398 N

This is the limiting static friction that shall be developed between the surfaces of contact. As the horizontal component P (i.e. 200 cos 60^o = 100 N) is less than 113.398 N, the block A shall not move if the coefficient of friction is 0.5.

But, if the limiting static friction is less than the horizontal component of P, then the block A will move.

∴ F_A  ≤ 100 N

∴ F_A ( max )  = 100

∴ μ . N _A = 100

∴          μ _{( max )} = 100 / N_A

=          100 /226.795

=        0.4409 N

Thus, if the coefficient of friction is 0.4409 or less, then the block A shall move under the given conditions.