Name: Necessity of Friction
Uploaded: 2019-07-19
Duration: 4 min 40 s
Description: This video enables students to learn about friction, friction used in routine activities, loss of energy in the form of heat due to friction.

Question

A body of weight

W

rests on a rough inclined plane and a force

P

acting at an angle

α

with the inclined plane just prevents the body from sliding down. If the inclined plane makes an angle

ϕ

with the horizontal, prove that

P = W

.

\frac{\sin (θ - λ)}{\cos (α + λ)} [λ = a n g l e o f f r i c t i o n]

.

Accepted Answer

Given a body of weight $W$ placed on a plane inclined with angle $ϕ$ with the horizontal. A force $F$ is applied to the body by pulling by a rope making an angle $α$ with the inclined plane

The forces applied on the body are:

(1) The weight $\frac{\sin (θ - λ)}{\cos (α + λ)} [λ = a n g l e o f f r i c t i o n]$ vertically downward.

(2) The normal reaction $R = mg \cos ϕ$ , perpendicular to the plane.

(3) the force of friction along the plane.

Refer to the FBD shown above about the situation, we have the following relation at the equilibrium:

Along the $X$ - axis $m g \sin ϕ + F_{s} = F \cos α$ .....................(1)

Along $Y$ - axis $R + F \sin α = m g \cos ϕ$ ........................(2)

Where $F_{s} = μR = μ (mg cosϕ - F sinα)$ .

$mg \sin ϕ + μmg \cos ϕ - μF sinα = Fcos α$ .

Or $mg (\sin ϕ + μ \cos ϕ) = F (\cos α + μ \sin α)$ .

Therefore $F = \frac{mg \sin ϕ + μ mg cosϕ}{\cos α + μsin α}$ ................................(3)

as $λ$ = angle of friction hence $μ = \tan λ = \frac{\sin λ}{cosλ}$ .

Substituting for $μ$ we get , $F = \frac{mg (\sin ϕ + \frac{sinλ}{cosλ} . cosϕ)}{\cos α + \frac{\sin λ}{\cos λ} . \sin α} = \frac{mg cosλ (\sin ϕ cosλ + \sin λcosϕ)}{\cos λ (\cos λcosα + sinλsinα)}$ .

On simplifying , we get ,

$F = W \frac{Sin (ϕ - λ)}{\cos (α + λ)}$ .

.

Important Questions on Equilibrium of Concurrent Forces and Friction

Learn and Prepare for any exam you want !

A body of weight W rests on a rough inclined plane and a force P acting at an angle α with the inclined plane just prevents the body from sliding down. If the inclined plane makes an angle ϕ with the horizontal, prove that P = W. sin θ-λcos α+λλ = angle of friction.

Important Questions on Equilibrium of Concurrent Forces and Friction

Learn and Prepare for any exam you want !