Question

A block of mass $2kg$ is pushed normally against a rough vertical wall with a force of $40N$, co-efficient of static friction being $0.5$. Another horizontal force of $15N$ is applied on the block in a direction parallel to the wall. (Will the block move? If yes, in which direction and with what minimum acceleration? If no, find the friction force exerted by the wall on the block.)

Answer 1

The force which may cause the tendency of motion or motion in the body is its own weight and the applied horizontal force of $15N$. The resultant of the forces.
$F=\sqrt{{20}^2+{15}^2}=25N$ in a direction ${tan}^{-1}\left(\frac{15}{20}\right)={37}^o$ with the vertical.
The friction will, by its very virtue of opposing the tendency of relative motion act in a direction opposite to the resultant force. Now, for the acceleration to be minimum, the minimum force requires$=F-\mu N$ (as, $\mu N$ is the maximum friction force)
$=25-0.5\times 40=5N$
Minimum acceleration is $\frac{5}{2}=2.5m/s^2$