8.15 Incline plane

Generally, the incline is considered to be fixed to the ground. If it is not fixed, then it is important to know the nature of friction between the incline and horizontal surface on which the incline is placed. If their interface is smooth, then any force on incline will accelerate the incline itself. Even just placing a block on the incline will move the incline. The horizontal component of normal force (N sinθ) applied by the block will accelerate the incline to the right with acceleration, ${\mathbf{a}}_I$ .

In that situation, the motion of block is taking place in the accelerated frame of the incline – not in the inertial frame of the ground. The acceleration of the block with respect to incline ( ${\mathbf{a}}_{\mathrm{BI}}$ ) is along the incline.

The acceleration of the block with respect to ground ( ${\mathbf{a}}_B$ ) is obtained from the relation for relative acceleration given as :

where ${\mathbf{a}}_I$ is the acceleration of incline with respect to ground and ${\mathbf{a}}_{\mathrm{BI}}$ is the acceleration of block with respect to incline. Now, evaluating the vector sum of the accelerations of the accelerations on the right hand side, the direction and magnitude of acceleration of the block with respect to ground is obtained.

However, if the contact between the incline and horizontal surface is rough, then the motion of incline will depend on whether force on the incline in horizontal direction (parallel to contact surface) is greater than the maximum static friction or not?

We need to be careful while applying Newton’s force laws for the case, where incline itself is accelerated. We should evaluate the situation as required either using ground reference or the accelerated reference of the incline – whichever is suitable for the situation in hand. We shall discuss this aspect of incline motion in details after studying motion in accelerated reference.

In this module, we shall, therefore, restrict our discussion to motion of a body on an incline plane, which is (a) stationary with respect to ground and (b) offers negligible friction to the motion of the body.

Motion on a smooth fixed incline

There are only two external forces on the block moving on a smooth incline plane. They are (a) normal force and (b) weight of the block.

There is no motion in the direction perpendicular to the incline. As such, forces in that direction form a balance force system. The normal force equals component of weight in opposite direction.

On the other hand, there is only component of weight along the incline in downward direction that accelerates the block in that direction. It must clearly be understood that the block will not be at rest on a smooth incline, unless some additional force stops the motion. The acceleration along the incline is obtained as :