Final answer:
Using the equation for torque (τ = Iα) and the moment of inertia formula for a rectangle (I = m · r^2/3), the width of each section of the rotating door is calculated to be approximately 1.5 meters.
Step-by-step explanation:
To determine the width of each section of the rotating door, we can use the concept of torque and the rotational equation of motion. Torque (τ) is given by the product of the force (F) and the lever arm (r), which in this case is the width that we need to find. The door sections exhibit rotational motion, so we can use the second rotational law of motion, τ = Iα, where I is the moment of inertia and α is the angular acceleration.
To calculate the width of the door, we need to derive the moment of inertia for a single section, which for a rectangular section of a rotating door about an axis perpendicular to the door and through its center can be calculated using the formula for a rectangle: I = m · r2/3, where m is the mass of the section and r is the width we are solving for.
Given the mass (m) of 80 kg per section, a force (F) of 60 N, and an angular acceleration (α) of 0.460 rad/s2, we can start by finding the torque using the force and angular acceleration:
torque using the force and angular acceleration:
τ = F · r
τ = Iα
60 N · r = (80 kg · r2/3) · 0.460 rad/s2
From the above equation, we solve for r:
60 N · r = 26.4 kg · r2
r = √(60 N / 26.4 kg·s2)
r ≈ 1.5 meters
Therefore, the width of each section of the rotating door is approximately 1.5 meters.