What should be the coefficient of kinetic friction so that a ice cube kept on a slope inclined at 30 degrees does not slide down.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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An ice cube is kept on a slope inclined at 30 degrees. Let the mass of the cube be M. The weight of the cube is M*g. This can be divided into components parallel and perpendicular to the slope. The component of M*g parallel to the slope is M*g*sin 30, this is the force that accelerates the cube downwards. And the component of M*g perpendicular to the slope is M*g*cos 30. This is the normal force. The frictional force opposing the movement of the cube is M*g*cos 30*C, where C is the coefficient of kinetic friction.

If the cube does not slide down, M*g*cos 30*C = M*g*sin 30

=> tan 30  = C

=> C =  0.57735

The coefficient of kinetic friction has to be 0.57735 for the ice cube to not slide down the inclined slope.