Question

A force of 100 N needs to be applied parallel to a smooth inclined plane just to hold a body on it. The angle of inclination of the inclined plane is ${30}^{\circ }$. How much horizontal force needs to be applied to do the same?

• A. 50 N
• B. 87 N
• C. 100 N
• D. 115 N

Answer 1

The correct option is D

115 N

Lets draw the diagram and see what's happening.

Free Body Diagram of block

Body at equilibrium taking components along and perpendicular and equating net force to 0 we get

$\sum f_x=0$ $\sum f_y=0$

$\sum f_x=0$

$\Rightarrow$ $100-mgsin{30}^{\circ }=0$

$100=mg$

$\frac{mg}{2}=200$ ------------ (1)

New scene

A force is being applied horizontally and its holding block stationary (equilibrium) diagram

Free Body Diagram

Again the body is in equilibrium

$\sum f_x=0$ $\sum f_y=0$

$\sum f_x=0$

$\Rightarrow$ $Ncos{30}^{\circ }-mg=0$

$Ncos{30}^{\circ }=mg$

From equation (1)

$N\frac{\sqrt{3}}{2}=200$

$N=\frac{400}{\sqrt{3}}$ ------------------(2)

$\sum f_x=0$

$\Rightarrow F-Nsin{30}^{\circ }=0$

$F=Nsin{30}^{\circ }$

From equation (2)

$F=\frac{400}{\sqrt{3}}\times \frac{1}{2}=\frac{200}{\sqrt{3}}$

$F=115.4N$