JEE Main & Advanced Mathematics Statistics Limiting Equilibrium on an Inclined Plane

Let a body of weight W be on the point of sliding down a plane which is inclined at an angle \[\alpha \]  to the horizon. Let R be the normal reaction and \[\mu R\] be the limiting friction acting up the plane.

Thus, the body is in limiting equilibrium under the action of three forces : \[R,\,\,\mu R\] and W.

Resolving the forces along and perpendicular to the plane, we have \[\mu R=W\sin \alpha \text{ and }R=W\cos \alpha \]

\[\Rightarrow \frac{\mu R}{R}=\frac{W\sin \alpha }{W\cos \alpha }\Rightarrow \mu =\tan \alpha \] \[\Rightarrow \tan \lambda =\tan \alpha \Rightarrow \alpha =\lambda \]

Thus, if a body be on the point of sliding down an inclined plane under its own weight, the inclination of the plane is equal to the angle of the friction.

(1) Least force required to pull a body up an inclined rough plane : Let a body of weight W be at  point \[A,\,\,\alpha \] be the inclination of rough inclined plane to the horizontal and \[\lambda \] be the angle of friction. Let P be the force acting at an angle \[\theta \] with the plane required just to move body up the plane.

\[P=W\frac{\sin (\alpha +\lambda )}{\cos (\theta -\lambda )}\],       \[\left\{ \because \mu =\tan \lambda  \right\}\]

Clearly, the force P is least when \[\cos (\theta -\lambda )\]is maximum, i.e. when \[\cos (\theta -\lambda )=1\], i.e. \[\theta -\lambda =0\]or \[\theta =\lambda \]. The least value of P is \[W\sin (\alpha +\lambda )\]

(2) Least force required to pull a body down an inclined plane : Let a body of weight W be at the point A, a be the inclination of rough inclined plane to the horizontal and l be the angle of friction. Let P be the force acting an angle q with the plane, required just to move the body up the plane.

\[P=\frac{W\sin (\lambda -\alpha )}{\cos (\theta -\lambda )}\],        \[[\because \mu =\tan \lambda ]\]

Clearly, P is least when \[\cos (\theta -\lambda )\] is maximum, i.e. when \[\theta -\lambda =0\] or \[\theta =\lambda \]. The least value of P is W\[\sin (\lambda -\alpha )\].  

• If \[\alpha =\lambda \], then the body is in limiting equilibrium and is just on the point of moving downwards.

• If \[\alpha <\lambda \], then the least force required to move the body down the plane is \[W\sin (\lambda -\alpha )\].

• If \[\alpha =\lambda ,\,\alpha >\lambda \text{ or }\alpha <\lambda \], then the least force required to move the body up the plane is \[W\sin (\alpha +\lambda )\].

• If \[\alpha >\lambda ,\]then the body will move down the plane under the action of its weight and normal reaction.