Final answer:
The magnitudes of the forces on the ladder can be calculated using static equilibrium, which requires the sum of forces and torques to be zero. By setting up equations for horizontal forces, vertical forces, and torques with a pivot point at the base, the unknown normal and frictional forces acting on the ladder can be solved when a person is on it.
Step-by-step explanation:
To determine the magnitudes of the forces on a ladder at the top and bottom when a person is standing on it, we must consider the ladder and the person as a system and apply the principles of static equilibrium. Static equilibrium requires that both the net force and net torque about any pivot point be zero.
The weight of the person (W_person) creates a downward force, and this force can be represented by the person's mass (70.0 kg) times the acceleration due to gravity (9.8 m/s²). The ladder's weight (W_ladder) is the product of its mass (10.0 kg) and gravity as well. The ladder makes contact with the ground at the base and with the wall at its top. Due to the ladder's weight and the person's weight on it, there is a normal reaction force at the base of the ladder (N) pushing up and a horizontal static friction force (f) at the base that resists sliding. At the top of the ladder, there will be a horizontal force (F) exerted by the wall. If the surface at the top is frictionless, as stated, there is no vertical force at the top.
To find the magnitudes of the normal and friction forces (N and f) at the base of the ladder, you need to set up equations for the sum of the horizontal forces and the sum of the vertical forces, as well as sum of the torques being equal to zero. With the pivot point at the base of the ladder, only the weights and the person apply a torque because the normal and frictional forces both act at the pivot, meaning they do not produce a torque. Using these conditions, you can then solve for the unknown forces.