Slippery Slope

The film Touching the Void tells the true story of Joe Simpson and Simon Yates and their 1985 attempt to scale Siula Grande. While successful with the ascent, the steep descent they attempted got them in serious trouble.

Simpson falls and breaks his leg part-way down the mountain, and Yates attempts to lower him down a particularly dangerous slope by means of a rope since he cannot walk. The slope is inclined at źŴ to the horizontal, and the coefficient of friction between Joe Simpson and the snowfield is ŴŻ. Joe is being held in equilibrium by a rope at an angle of Ź from the slope. Draw a force diagram to show all forces acting on Joe.

Assuming Joe (and his kit) weighs ŵŴŴ˫˧ , find the minimum tension in the rope.

What is the largest tension that could be applied to the rope?

Slippery Slope Solutions

Assuming Joe (and his kit) weighs ŵŴŴ ˫˧ , find the minimum tension in the ro pe.

Minimum tension occurs when friction is at maximum: ˘ ŴŻ˞

Resolving parallel to the slope: ˘ -ˠU[_Ź ŵŴŴ˧_ZźŴ ŹŴ˧ŷ ŴŻ˞ - ˠU[_Ź ŹŴ˧ŷ

Resolving perpendicular to the slope: ˞-ˠ_ZŹ ŵŴŴ˧U[_źŴ ŹŴ˧

'" %!" =$G ' Solving simultaneously: ˞ ŹŴ ˧ . ˠ _Z Ź ˤ ˞ "&

ŴŻ'ŹŴ˧ . ˠ _Z Ź( ŹŴ˧ŷ .ˠU[_Ź ŷŹ˧ .ŴŻˠ_ZŹ ŹŴ˧ŷ . ˠ U[_ Ź

)ŹŴ ŷ . ŷŹ *˧ ˠ'U[_Ź.ŴŻ_ZŹ( )ŹŴŷ . ŷŹ *˧ ˠ +,-./Ϳ1΃ 3 U[_Ź. ŴŻ_ZŹ

What is the largest tension that could be applied to the rope?

At maximum tension, friction will be maximum, but acting down the slope: ˘ ŴŻ˞

Resolving parallel to the slope: ˠU[_Ź ŵŴŴ˧_ZźŴ-˘ ŹŴ ŷ˧ - ŴŻ˞

Resolving perpendicular to the slope: ˞-ˠ_ZŹ ŵŴŴ˧U[_źŴ ŹŴ˧

" =$G '!'" % Solving simultaneously: ˞ ŹŴ˧ . ˠ_ZŹˤ˞ "&

ˠ U[_ Ź . ŹŴ ŷ˧ ŹŴ ˧ . ˠ _Z Ź ŴŻ'ŹŴ ˧ . ˠ _Z Ź( ˠ U[_ Ź . ŹŴ ŷ˧ ŴŻ

ŷŹ˧.ŴŻˠ_ZŹ ˠU[_Ź.ŹŴŷ˧

)ŹŴ ŷ - ŷŹ *˧ ˠ'U[_Ź-ŴŻ_ZŹ( )ŹŴŷ - ŷŹ *˧ ˠ --14./Ϳ1΃ 3 U[_Ź-ŴŻ_ZŹ