10.6 Induced motion on rough incline plane

The natural tendency of a block is to move along the incline in downward direction under the “pull” of gravity. Motion of the block, however, can be induced either up or down along the incline by applying additional force other than three forces (weight, normal force and friction), which always operate on the block.

We will study the effect of an additional external force on the motion of a block on the rough incline plane. An additional external force has multiple implications on the state of motion of the block. Some of the important implications are :

• External force can change the normal force (N) and hence maximum static friction for a given pair of surfaces. As a consequence, the force requirement parallel to contact surface to initiate motion will also change.
• External force can change the direction of motion with respect to the incline plane. The block can move either up or down (or the tendency to move up or down) the incline, depending on the direction of component of net external force at the contact surface.
• Friction is opposite to the direction of the motion (or tendency of motion) of the block with respect to incline.

Most importantly, external force can overcome normal limitations to motion, resulting from requirement to exceed friction. It is so because the limitations are there, because of the fact that force due to gravity (mg), which is responsible for motion, is a constant for a given mass of the block. Such is not the limitation with the applied external force (F). It can have any magnitude and direction.

Direction of friction

The externally unaided force system of three forces (weight, normal force and friction) causes downward motion (or induces tendency for motion in downward direction). On the other hand, the force system that involves additional external force (F) may have component of external forces parallel to contact in either up or down direction. As such, we need to analyze the force system without friction to decide the direction of friction. The figure below highlights this aspect showing question mark against the direction of friction.

In short, there are two possibilities depending on the component of net external force (excluding friction) at the contact. Let ( $F_{\mathrm{||}}$ denotes the component of net external force (excluding friction) parallel to contact surface. Then :

• If $F_{\mathrm{||}}$ <0, then the block will either have the tendency to move down or there will be downward motion. The friction will then act in the upward direction.
• If $F_{\mathrm{||}}$ >0, then the block will either have the tendency to move up or there will be upward motion. The friction will then act in the downward direction.

Nature of external force and direction of friction

The external force can be distinctly different in one important aspect. The component of external force parallel to the contact surface can either be downward or upward. This distinction of the additional external force provides us with a direct method for determining the direction of friction.