The last lecture was one equation short. What remains is the balance of moments, so we write it.
Moments may be measured about any point. If the body is balanced, the sum is zero wherever it is taken, so we are free to choose whichever point suits us.
The point to choose is the lower end of the rod. Take it as the centre of rotation and the arms of both floor forces, and , become zero. A force applied at the very centre of rotation does nothing to turn the body about it. Those two arrows are dropped from the figure to show that they are gone.
What we want is , so we chose the point at which everything but disappears. Pick the point well and the forces you have no interest in never enter the equation. This is the single most useful habit in working with moments.
Measure the arms of the two forces that are left. An arm is the distance from the centre of rotation to the line of action of the force.
points horizontally, so its arm is the vertical distance from the lower end, which is the height of the upper end of the rod, .
points vertically, so its arm is the horizontal distance from the lower end. The centre of gravity is at the middle of the rod, so that is .
The two turn the rod in opposite senses. turns it toward upright, turns it toward flat. They are in balance, so the products of force and arm are equal, giving .
Solve that for . Both sides carry , so dividing removes it. And is , so it gathers into .
The length of the rod has gone. Long rod or short, if it leans at the same angle it pushes the wall with the same force. Lengthening the rod lengthens the arm, but it moves the centre of gravity out by just as much, and the two cancel.
Swing the angle in the figure. Lay the rod flatter and falls, so grows. Stand it up and shrinks. The obvious fact that leaning harder on a wall pushes it harder is written directly into the equation.
That is the third equation. , and can now all be written in terms of and .
Finally, write the condition for not slipping: the friction at the floor must not exceed its limit, that is, .
Put the results in. With and , this becomes . Both sides carry , so dividing removes it, and is left.
Both the length and the weight have gone. What remains is the angle and the coefficient of friction at the floor. Long ladder or short, with someone on it or without, the angle of slipping is the same. If is then , which is to say the ladder must stand steeper than .
The figure starts upright and lays the rod down. The friction arrow grows and stops when it reaches the marker for the limit. That is the critical angle. The warning against setting a ladder at too shallow an angle is this inequality.