Problem: The penguin's mass is 50 kg and the ladder's is 22 kg. If the penguin ascends the ladder two-thirds of its length before it slips, what is the frictional force exerted by the ground if the wall is essentially frictionless? The ladder is placed at an angle of 70o from the ground.
Solution: It is not so unrealistic to assume that the wall is frictionless. When you ascend a ladder, do you rely on friction between the ladder and the wall? First, here's a pictorial representation of the problem:
Next I draw the free body diagram, where Ay is the normal force from the ground and Bx is the wall's normal force. Ax is the ground's frictional force.
Finally, I write the equations and substitute:
But these equations don't yield Ax and could have been skipped. It is the torque equation that yields the frictional force:
Point C was a very useful point about which to take torques, since Ay and Bx pass through it (zero moment arms and hence zero torques). That eliminates two unknowns from the torque equation. It is instructive to consider alternative points about which to take torques, and you should try others, making sure that you get the same answers.
In problems like this, if you don't have one sine function and the rest cosines (or vice versa), you should check the work very carefully - there's probably a mistake! Make sure that you can see how I have determined the moment arm for each case.