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Solve the following :

Question:

A block of mass $2 \mathrm{~kg}$ is pushed against a rough vertical wall with a force of $40 \mathrm{~N}$, coefficient of static friction being $0.5$. Another horizontal force of $15 \mathrm{~N}$, is applied on the block in a direction parallel to the wall. Will the block move? If yes, in which direction? If no, find the frictional force exerted by the wall on the block.

Solution:

Net driving force on the block

$F_{\text {driving }}=\sqrt{(m g)^{2}+(15)^{2}}$

Limiting friction force $=\mu R=0.5 \times 40=20 \mathrm{~N}$

$\approx F_{\text {driving }}>f f_{(\mathrm{lm})}$ block will move.

For acceleration,

$F_{\text {Net }}=25-20$

$m \times a=5$

$2 \times a=5$

$a=2.5 \mathrm{~m} / \mathrm{s}^{2}$

Direction w.r.t $15 \mathrm{~N}$ force

$\tan \phi=\frac{20}{15}=\frac{4}{3}$

$\phi=53^{\circ}$ with $15 \mathrm{~N}$ force.