Name: Two Block System
Uploaded: 2019-07-19
Duration: 6 min 20 s
Description: Two blocks placed on a surface exerts a force between them and also with the surface. Let us discuss how these forces can be solved numerically in this video.

Question

A block of mass

m

rests on a horizontal floor with which it has a coefficient of static friction

μ

. It is desired to make the body move by applying the minimum possible force

F

. Find the magnitude of

F

and the direction in which it has to be applied.

Accepted Answer

Suppose the force, $F$ is applied at an angle, $θ$ with the horizontal as shown in the adjoining figure. For vertical equilibrium,

$R + F \sin θ = m g$

$\Rightarrow R = m g - F \sin θ . . . (i)$

While for horizontal motion

$F c o s θ \geq f_{L}$ or $F c o s θ \geq μ R . . . (ii)$

From equations $(i)$ and $(ii),$ we get;

$F \cos θ \geq μ (m g - F \sin θ)$

or $F \geq \frac{μ m g}{(\cos θ + μ \sin θ)}$

For the force $F$ to be minimum $(c o s θ + μ s i n θ)$ must be maximum,

i.e., $\frac{d}{d θ} (\cos θ + μ \sin θ) = 0$

$\Rightarrow - \sin θ + μ \cos θ = 0$

i.e., $\tan θ = μ$
$\Rightarrow θ = \tan^{- 1} (μ)$

$∴ s i n θ = \frac{μ}{\sqrt{1 + μ^{2}}}$ and $c o s θ = \frac{1}{\sqrt{1 + μ^{2}}}$

$∴ F ⩾ \frac{μ m g}{\frac{1}{\sqrt{1 + μ^{2}}} + \frac{μ^{2}}{\sqrt{1 + μ^{2}}}}$

$∴ F_{\min} = \frac{μ m g}{\sqrt{1 + μ^{2}}}$

A block of mass $m$ rests on a horizontal floor with which it has a coefficient of static friction $μ$ . It is desired to make the body move by applying the minimum possible force $F$ . Find the magnitude of $F$ and the direction in which it has to be applied.

Important Questions on Friction

Learn and Prepare for any exam you want !

A block of mass m rests on a horizontal floor with which it has a coefficient of static friction μ. It is desired to make the body move by applying the minimum possible force F. Find the magnitude of F and the direction in which it has to be applied.

Important Questions on Friction

Learn and Prepare for any exam you want !

A block of mass $m$ rests on a horizontal floor with which it has a coefficient of static friction $μ$ . It is desired to make the body move by applying the minimum possible force $F$ . Find the magnitude of $F$ and the direction in which it has to be applied.