A uniform ladder 10.0 m long is leaning against a frictionless wall at an angle of 60.0° above the horizontal. The weight of the ladder is 18.4 lb. A 60.8-lb boy climbs 4.18 m up the ladder. What is the magnitude of the frictional force exerted on the ladder by the floor?
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To find the magnitude of the frictional force exerted on the ladder by the floor, we need to consider the forces acting on the ladder.
Let's begin by finding the forces acting on the ladder. There are three forces involved: the weight of the ladder, the weight of the boy, and the normal force exerted by the floor.
1. Weight of the ladder: The weight of the ladder can be calculated using the formula: weight = mass x acceleration due to gravity. Given that the weight of the ladder is 18.4 lb, we can convert it into mass by dividing it by the acceleration due to gravity (g ≈ 9.8 m/s^2) using the formula: mass = weight / acceleration due to gravity.
weight_of_ladder = 18.4 lb
mass_of_ladder = weight_of_ladder / acceleration due to gravity
2. Weight of the boy: Similarly, we can find the weight of the boy by dividing his weight (60.8 lb) by the acceleration due to gravity.
weight_of_boy = 60.8 lb
mass_of_boy = weight_of_boy / acceleration due to gravity
3. Normal force exerted by the floor: The normal force is the force exerted perpendicular to the surface of contact, which balances the weight of the ladder and the boy. Since the ladder is in contact with the floor, the normal force exerted by the floor balances the vertical component of the weight of the ladder and the boy.
To calculate the normal force, we need to consider the vertical components of the weight of the ladder and the boy. The vertical component can be calculated using the formula: vertical_component = weight x cos(angle of inclination).
vertical_component_of_weight_of_ladder = weight_of_ladder x cos(angle of inclination)
vertical_component_of_weight_of_boy = weight_of_boy x cos(angle of inclination)
Since the ladder and the boy are in equilibrium (not accelerating vertically), the sum of the vertical components of their weights should be equal to the normal force exerted by the floor.
normal_force = vertical_component_of_weight_of_ladder + vertical_component_of_weight_of_boy
Finally, the frictional force exerted on the ladder by the floor is equal in magnitude but opposite in direction to the horizontal component of the weight of the ladder and the boy.
frictional_force = weight_of_ladder x sin(angle of inclination) + weight_of_boy x sin(angle of inclination)
To find the magnitude of the frictional force, you can plug in the values and calculate it using the above equations.