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The mathematics of a rope or cable passing over a rough surface, focusing on the forces acting in the tangential and normal directions. The text derives equations for tension and angle changes, and discusses the coefficient of friction's impact on these relationships. Students will work problems related to determining tension required for a one-ton weight and the friction coefficient's dependence on tension.

Typology: Lab Reports

Pre 2010

1 / 2

Download Cable Friction and Tension on Rough Surfaces and more Lab Reports Calculus in PDF only on Docsity! Lab 8. Cable in Contact with a Rough Surface. Here we consider the mathematics of a rope or cable passing over a rough surface. You may have already come across the phenomenon of wrapping a rope around a tree or a post, for example, and producing a situation in which huge loads can be held. You may have seen tree cutters use this to lower branches. Consider the situation as depicted in the sketch. We will now balance the forces acting on a small piece of the cable ∆s. First we balance the forces in the tangential direction: T F s T T+ = +∆ ∆ ∆( ) cos( )θ . If we use the tangent line approximation for cos ( )∆θ , which is simply cos( )∆θ ≈ 1, we have, T F s T T+ ≈ +∆ ∆( ). Thus on letting ∆s→0, we derive, (i) dT ds F= . Secondly we balance the forces in the direction perpendicular to the tangent, that is to say along the normal, N s T T∆ ∆ ∆θ= +( )sin( ). If we use the tangent line approximation for sin ( )∆θ , which is sin( )∆ ∆θ θ≈ , we have, ∆θ ∆θ T , F T+∆T N