- The tension in the rope is approximately 220.25 N.
- The magnitude of the normal force that the floor exerts on the ladder is approximately 504.125 N.
Given:
- Mass of the ladder (m1) = 15 kg
- Mass of the child (m2) = 25 kg
- Distance of the child from the end X of the ladder (d) = 3/4 * length of the ladder = 3/4 * 2.0 m = 1.5 m
- Angle (θ) = 30 degrees
- Acceleration due to gravity (g) = 9.8 m/s²
We can start by calculating the forces due to the weight of the ladder and the child:
Weight of the ladder (W1) = m1 * g = 15 kg * 9.8 m/s² = 147 N
Weight of the child (W2) = m2 * g = 25 kg * 9.8 m/s² = 245 N
The ladder is in rotational equilibrium, so the sum of the torques about any point is zero. Let’s choose the point where the ladder touches the ground (point X) as the pivot point. The torques due to the weight of the ladder, the weight of the child, and the tension in the rope then have to balance out:
Sum of torques = Torque_due_to_W1 + Torque_due_to_W2 - Torque_due_to_Tension = 0
Given that the torque due to a force is given by the product of the force and the distance from the pivot point, and that the weight of the ladder acts at its midpoint (1.0 m from the pivot point), the weight of the child acts 1.5 m from the pivot point, and the tension acts 2.0 m from the pivot point, we can write:
147 N * 1.0 m + 245 N * 1.5 m - Tension * 2.0 m = 0
Tension = (147 N * 1.0 m + 245 N * 1.5 m) / 2.0 m = 220.25 N
So, the tension in the rope is approximately 220.25 N.
Next, let’s calculate the normal force exerted by the floor on the ladder. The ladder is in translational equilibrium, so the sum of the vertical forces is zero. The vertical forces include the normal force, the vertical components of the weight of the ladder and the child, and the vertical component of the tension in the rope:
Sum of vertical forces = Normal_force - W1 - W2 - Tension * sin(θ) = 0
Normal_force = W1 + W2 + Tension * sin(θ) = 147 N + 245 N + 220.25 N * sin(30 degrees) = 504.125 N
So, the magnitude of the normal force that the floor exerts on the ladder is approximately 504.125 N.