First slide

Friction

Question

A block of mass m is lying on a horizontal surface of coefficient of friction $μ$ . A force F is applied to the block at an angle $θ$ with the horizontal. The block will move with a minimum force F if

Moderate

A

$μ = \tan θ$
B

$μ = \cot θ$
C

$μ = \sin θ$
D

$μ = \cos θ$

Solution

Refer to diagram,
Vertical component of F is F sin $θ$ and the horizontal component is F cos $θ$ . Thus
$R + F \sin θ = mg$ or    $R = mg - F \sin θ$

Frictional force   $μR = μ (mg - F \sin θ)$ .
Also $μ (mg - F \sin θ) = F \cos θ$
or $F = \frac{μmg}{(μ \sin θ + \cos θ)} (i)$
F will be minimum if the denominator is maximum, i.e., if $\frac{d}{dθ} (μ \sin θ + \cos θ) = 0$
or $μcos θ - \sin θ = 0$ or $μ = \tan θ$

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