Equilibrium in 2D - Ladder Problems Practice Problems

A painter places a 13 m long regular ladder against a vertical wall with negligible friction. The ladder's lower end is 5 m from the wall. Take the weight of the ladder to be 250 N and the coefficient of static friction between the floor and the ladder's foot to be 0.65. If a painter weighing 650 N climbs gently up the ladder, determine the maximum friction force that the floor can apply on the ladder's lower end.

An extendable pole that is 4m in length is placed against a smooth, vertical surface. The static friction coefficient between the pole and the ground is 0.50. Calculate the smallest angle the pole can form with the ground to stay in position without sliding.

What is the minimum angle the rods can make to the surface and still not fall when two identical rods are set up symmetrically against each other on a surface with a coefficient of friction 0.51?

Consider a situation where a ladder is resting on a smooth vertical surface, forming an angle with the ground. The ladder is of uniform density and has a mass of 10.0 kg. A person with a mass of 60.0 kg stands on the ladder. As the person walks towards the higher end of the ladder, it starts slipping when the person is 60% of the way from the lower end to the higher end along the length of the ladder. Determine the coefficient of static friction(μ) between the ladder and the ground.

A uniform beam of mass $m$ and length $L$ leans against a rough wall as shown below. The coefficients of static friction between the beam and the floor, and between the beam and the wall, are $\mu_{f}=0.50$ and $\mu_{w}=0.30$ , respectively. Assume the beam is on the verge of slipping and stability is ensured if $\theta \geq \theta_{\min }$, given by

Calculate the true value of $\theta_{\min }$. Then, assuming the wall is frictionless ($\mu_{w}=0$), the expected value of $\theta _{\min }\text{ is }45^{\circ }$. Determine the percentage error between the true value and the frictionless wall approximation.