**Branch :**First Year-Engineering Syllabus

**Subject :**Elements of Mechanical Engineering

## Belt Friction

** Belt Friction: **If a rope wrapped around a rough post is subjected to a large force on one of its ends, a small force on the other end may be

fig..(1)

able to prevent the rope from slipping. In Fig. 9.8a, the rope is wrapped around the post at an angle α. It is assumed that the force S_{2} applied to the left end of the rope is larger than the force S_{1} exerted on the right end. In order to establish a relation between these forces, we draw the free-body diagram shown in Fig. 1b and apply the equilibrium conditions to an element of the rope with length ds. In this context, we take into account that the tension is changing by the infinitesimal force dS along ds. Since S_{2 }> S_{1} holds, the rope would slip to the left without friction; the static friction force dH is therefore oriented to the right. The equilibrium conditions can be formulated as follows:

Since dϕ is infinitesimally small, we obtain cos (dϕ/2) ≈ 1 and sin (dϕ/2) ≈ dϕ/2; furthermore, the higher order term dS(dϕ/2) is small and can be neglected in the following. Therefore, the above relations simplify to

..........................eq(1)

Obviously, the three unknowns H, N, and S cannot be determined from these two equations: the system is statically indeterminate. Therefore only the limiting friction case is considered, i.e., when slippage of the rope is impending. In this case gives

Applying (1) yields

.........................eq(2)

This formula for belt friction is commonly named after Leonhard Euler (1707–1783) or Johann Albert Eytelwein (1764–1848) . If, in contrast to the initial assumption, S_{1} > S_{2} holds, one simply has to exchange the subscripts to obtain

......eq(3)

For a given S_{1}, the system is in equilibrium provided that the value of S_{2} remains within the limits given in (2) and (3):

......................eq(3)

The rope slips to the right if S_{2} < S_{1} e^{−μ0}, whereas it slips to the left for S_{2 }> S_{1} e^{μ0}. The following numerical example provides a sense of the ratio between the two forces. We assume the rope to be wrapped ntimes around the post; the coefficient of static friction is given by μ0 = 0.3 ≈ 1/π. In this case we obtain