Question

A particle of mass 2kg is placed on a rough plane inclined at an angle 35° to the horizontal.
The coefficient of friction is 2
3
. Find the least force parallel to the plane that is required
(i) to hold the particle at rest
(ii) to make the particle slide up the plane.

Answer 1

$F_{fr}=\mu N$

$$\vec{R}=m\vec{g}+\vec{N}+\vec{F_{fr}}+\vec{F}$$

(i) According to second Newton's Law

$$\vec{R}=0$$

$$m\vec{g}+\vec{N}+\vec{F_{fr1}}+\vec{F}=0$$

$$Ox: -\mu N-F+mg\sin\theta=0$$

$$Oy:N-mg\cos\theta=0$$

$$F=mg\sin\theta-\mu mg\cos\theta=mg(\sin\theta-\mu \cos\theta)$$

$$F=2\ kg\cdot9.81\ m/s^2(\sin35\degree-0.23\cos35\degree)\approx7.557 N$$

(ii) According to second Newton's Law

$$\vec{R}=0$$

$$m\vec{g}+\vec{N}+\vec{F_{fr2}}+\vec{F}=0$$

$$Ox: \mu N-F+mg\sin\theta=0$$

$$Oy:N-mg\cos\theta=0$$

$$F=mg\sin\theta+\mu mg\cos\theta=mg(\sin\theta+\mu \cos\theta)$$

$$F=2\ kg\cdot9.81\ m/s^2(\sin35\degree+0.23\cos35\degree)\approx14.950 N$$