10.4 Motion on rough incline plane

Motion of a block on a rough incline plane is the interplay of different force types and the characterizing features of the incline surface. An incline is a plane surface, whose one end is raised. The raised surface forms an angle “θ” with the horizontal. The block or a body placed on the surface is acted upon by gravity and contact forces (normal and friction forces), besides other forces. In this module, we shall restrict our consideration to force due to gravity and contact forces and shall consider other external force in subsequent module.

Study of motion of a block on an incline is important from the point of view of the measurement of the coefficients of friction.

Static friction and incline

In this section, we consider the motion of a block placed on a stationary incline i.e. incline itself does not move on the horizontal surface. At present, we do not consider any additional force. The forces on the block are (i) weight of the block, mg, (ii) Normal force, N, and (iii) friction, $F_F$ ( $f_s$ , $F_s$ or $F_k$ ).

For the sake of illustration, we treat the incline as a plane surface, which can be lifted at one end with certain mechanical arrangement so that the incline angle “θ” with horizontal direction can be varied. This set up enables us to change the gravitational pull on the block in the direction of incline surface and work with incline of different angles for the study of motion on the incline.

The forces, in the direction normal to the incline plane, are balanced as there is no motion in that direction. On the other hand, the component of weight parallel to contact surface, mg sinθ, tends to initiate motion of the block along the incline plane. For small value of "θ", the component of force parallel to contact surface is small and is counteracted by friction, acting in the opposite direction such that :

$$\begin{array}{l}{f}_s=mg\sin \theta \end{array}$$

Angle of repose

As we increase the angle, θ, the downward force along the incline increases. The friction, in response, also increases till it reaches the maximum value corresponding to limiting (maximum) static friction, $F_s$ . In that situation,

$$\begin{array}{l}{F}_s=mg\sin \phi \end{array}$$

where “φ” is the angle at which the block just starts to slide down. This angle is known as “angle of repose”.

Measurement of coefficient of friction

We can measure coefficient of friction between two surfaces by measuring angle of repose. The free body diagram of the block as superimposed over the block-incline system corresponding to the maximum static friction is shown in the figure.

$$\begin{array}{l}\sum F_x=mg\sin \phi -F_s=0\\ \Rightarrow mg\sin \phi =F_s\end{array}$$

and

$$\begin{array}{l}\sum F_y=N-mg\cos \phi =0\\ \Rightarrow mg\cos \phi =N\end{array}$$

Combining two equations, we have :

$$\begin{array}{l}\Rightarrow \tan \phi =\frac{F_s}{N}\end{array}$$

However, we know that coefficient of static friction is given by :

$$\begin{array}{l}\Rightarrow {\mu }_s=\frac{F_s}{N}\end{array}$$

Hence,

$$\begin{array}{l}\Rightarrow {\mu }_s=\tan \phi \end{array}$$