# 12.13.1. Equilibrium of an Original Body

A uniform ladder of weight N is resting with its upper end against a smooth vertical wall and with its lower end against a rough horizontal ground, where the coefficient of friction between the ladder and the ground is . If a force of magnitude N acts on the lower end of the ladder trying to move it away from the wall, where the force makes an angle of with the horizontal, the ladder is about to slide. Determine the tangent of the angle that the ladder makes with the horizontal ground.

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### Example

A uniform ladder of weight N is resting with its upper end against a smooth vertical wall and with its lower end against a rough horizontal ground, where the coefficient of friction between the ladder and the ground is . If a force of magnitude N acts on the lower end of the ladder trying to move it away from the wall, where the force makes an angle of with the horizontal, the ladder is about to slide. Determine the tangent of the angle that the ladder makes with the horizontal ground.

### Solution

Resolve the forces horizontally.

Now resolve the forces vertically.

Find by substituting the value of in the first equation.

The sum of the algebraic measure of moments of the forces around is zero.

Divide by .

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