First slide

Static friction

Question

A body of mass m is kept on a rough horizontal surface (coefficient of friction = $μ$ ). Horizontal force is applied on the body, but it does not move. The resultant of normal reaction and the frictional force acting on the object is given F, where F is [NEET (Odisha) 2019]

Moderate

A

$|F| = m g + μ m g$
B

$|F| = μ m g$
C

$|F| \leq m g \sqrt{1 + μ^{2}}$
D

$|F| = m g$

Solution

The situation can be drawn as
The frictional force, f = $μ$ N = $μ$ mg ( $∵$ N=mg)
From free body diagram, the resultant force is
$| F | = \sqrt{N^{2} + f^{2}}$
$\leq$ $\sqrt{{(m g)}^{2} + {(μ m g)}^{2}} = m g \sqrt{1 + μ^{2}}$
This is the minimum force required to move the object. But as the body is not moving.
$∴$ $|F| \leq m g \sqrt{1 + μ^{2}}$

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