A uniform staircase weighing 20 kg is leaning against a smooth vertical wall, making an angle of 30 ° with it.

A uniform staircase weighing 20 kg is leaning against a smooth vertical wall, making an angle of 30 ° with it. The floor is rough. What is the modulus of the reaction force acting on the upper end of the ladder?

Let’s choose the coordinate system associated with the staircase (the X axis is directed along the staircase). Then, according to Newton’s second law in projection onto the Y-axis, we obtain:
N-mg * cos (90-α) = ma since the staircase is at rest a = 0
N-mgcos (90-α) = 0
N = mgcos (90-α) = 20 * 9.8 * 1/2 = 98 n