Final answer:
To find the magnitudes of the forces on the ladder, we use the equilibrium conditions for torques and forces. The force at the top can be found by equating the sum of torques due to the person and ladder's weight to the top torque and then solving for the normal force at the bottom.
Step-by-step explanation:
The question involves calculating the magnitudes of the forces exerted on a ladder at different points due to the weight of a person and the ladder itself. The ladder is placed against a frictionless surface, making it a statics problem that can be solved using equilibrium conditions for torques and forces. We'll use the weight of the person (mass = 70.0 kg) and the ladder (mass = 10.0 kg), the distances given, and assume standard gravity (9.8 m/s2) to find the forces at the top and bottom of the ladder.
We first calculate the torques about the base of the ladder to find the force at the top. The torque due to the person's weight is (70 kg × 9.8 m/s2) × 3 m, and the torque due to the ladder's weight is (10 kg × 9.8 m/s2) × 2 m. The sum of these torques must equal the torque due to the force at the top, which acts at a distance of 6 m from the base. Solving for the force at the top and applying Newton's second law in the vertical direction will give us the normal force at the bottom of the ladder. The friction force is zero since the gutter is frictionless.