10.6 Induced motion on rough incline plane  (Page 2/3)

We know that block has a tendency to move down on an incline due to the component of weight parallel to incline in the downward direction. Hence, if the component of additional external force is downward, then the induced motion has downward tendency and friction in upward direction. See the two figures below and orientations of the external force in each case. In either case, component of external force parallel to incline is directed downward. As such, the block is induced to move down.

Such clarity with respect to direction of friction and hence that of the direction of motion is not there, if the component of additional external force parallel to incline is in upward direction. In this case, the sum of the components parallel to incline may be less than or equal to or greater than the maximum static friction. In such situations as shown here in the figure, we need to analyze net force parallel to the contact surface to determine the direction of motion and then that of friction.

Analysis strategy

Force analysis for induced motion on incline plane will comprise of following steps :

• Identify the characteristics of applied external force to know the direction of its component along the incline.
• If the component of external force is down the incline, then motion is induced in downward direction and friction in upward direction.
• If the component of external force is up the incline, then analyze the external forces (excluding friction) to know the direction of component of net external forces parallel to the incline. Depending on the direction component of net external forces parallel to the incline, determine the direction of friction .
• After fixing the direction of friction, evaluate the state of friction (static friction, maximum static friction or kinetic friction) from the given set of information.
• Carry out force analysis along two mutually perpendicular directions, suited to the situation in hand.

Thus, sequential order of the analysis for induced motion on incline plane is as follows :

Additional force characterization -->Direction of motion -->Direction of friction -->Friction state -->Force analysis

Force analysis

In this section, we shall work with different states of friction (static friction, maximum static friction or kinetic friction) for applied additional external force on the block.

Case 1 : friction equal to kinetic friction

In this case, the block moves on the incline. Net force parallel to contact surface is greater than maximum static friction and friction is equal to kinetic friction.

$$\begin{array}{l}{F}_F=F_k={\mu }_{k}N\end{array}$$

Problem 1 : A block of 5 kg is pulled by a force 10 N making an angle 30° with the incline of angle 45°. If coefficients of static and kinetic frictions are 0.55 and 0.53, then determine friction between block and incline.

Solution : We observe here that component of additional external force has its component up the incline. We are, therefore, required to carry out force analysis without friction to ascertain the direction of motion. In the figure below, the free body diagram is superimposed on the block and incline system.