Final answer:
To find the tension in the cable, we need to apply the torque equation. By considering the weight of the plank and resolving the tension in the cable into its components, we can determine the magnitude of the tension. The tension in the cable is approximately 142.88 N.
Step-by-step explanation:
To find the tension in the cable, we need to consider the torque acting on the plank.
The weight of the plank acts at its center of mass, midway along its length, and can be considered as a single force of magnitude 47.0 kg * 9.8 m/s^2 = 461.6 N acting vertically downward.
The tension in the cable can be resolved into two components: one horizontal component acting parallel to the plank, and one vertical component acting upward to balance the weight of the plank.
By applying the torque equation, we can determine the magnitude of the tension in the cable.
The torque equation is given by:
τ = F * d * sin(θ)
Where τ is the torque, F is the force, d is the perpendicular distance from the pivot point to the line of action of the force, and θ is the angle between the force and the line connecting the pivot point and the point where the force is applied.
In this case, the torque equation becomes:
τ = (T * L * sin(θ)) - (W * (L/2) * sin(90 °))
Where T is the tension in the cable and W is the weight of the plank. Since the plank is held level, the torque must be zero, so we can solve for T:
T = W * (L/2) * sin(90 °) / (L * sin(θ))
Plugging in the given values, we can find the tension in the cable:
T = (461.6 N * (11.0 m/2) * sin(90 °)) / (11.0 m * sin(72 °))
T = 461.6 N / 11.0 * sin(72 °)
T ≈ 142.88 N