a) A rigid body is in mechanical equilibrium. State the two conditions that must be satisfied for this to be the case, both in words and in equations. [3]
A ladder of length 2L and mass M is positioned on level ground leaning against a wall such that the angle between the ladder and the horizontal is α. The coefficient of static friction between the ladder and the wall and between the ladder and the ground is μstatic = 0.65. The center of mass of the ladder is halfway along it.
(b) Draw a diagram indicating all the forces acting on the ladder, and state what each force represents. Also show the direction of the x- and y-axes you will use. (Hint: there are 2 forces acting at the top of the ladder, 2 forces acting on the bottom of the ladder and 1 force acting at its center.) [6]
(c) For the ladder to be in mechanical equilibrium:
(i) Write down equations for the total x- and y-components of the 5 forces acting on the ladder.
(ii) Consider torques about the center of the ladder. In which direction (into or out of the page) does the torque due to each of the 5 forces act?
(iii) Write down an equation for the sum of the torques about the center of mass of the ladder.
(iv) Use your equation for the torques to derive an expression for tan α in terms of the magnitudes of the forces acting. [9]
(d) (i) If the ladder is just on the point of slipping at both the upper and lower ends, what can you say about the pair of forces acting at each of the top and bottom of the ladder?
(ii) Hence use this information, with the information from (c)(i) and the expression you have derived in (c)(iv) to calculate the minimum angle that the ladder can form with the ground in order for it not to slip. (Hint: In the expression for tan α, you will need to write each of the forces in terms of one of the normal reaction forces, which then cancel out, leaving an expression for tan α in terms of μstatic only.) [6]