A rotating door is made from four rectangular sections, as indicated in the drawing. The mass of each section is 70 kg. A person pushes on the outer edge of one section with a force of F = 74 N that is directed perpendicular to the section. Determine the magnitude of the door's angular acceleration.
"in the drawing"
To determine the magnitude of the door's angular acceleration, we can use the formula for torque and rotational inertia.
The torque exerted on an object is given by the equation:
τ = I * α
Where:
τ is the torque,
I is the rotational inertia of the object, and
α is the angular acceleration.
The rotational inertia of an object depends on its shape and mass distribution. For a rectangular section rotating about an axis perpendicular to its face, the rotational inertia can be calculated using the formula:
I = (1/3) * m * h^2
Where:
m is the mass of the section, and
h is the height of the section.
Given that the mass of each section is 70 kg, and assuming the height of the section is known, we can calculate the rotational inertia.
Once we have the torque and rotational inertia, we can determine the magnitude of the angular acceleration using the formula:
α = τ / I
Let's assume the height of each section is h meters.
To determine the magnitude of the door's angular acceleration, we can use the equation for torque: τ = I * α, where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
In this case, the torque can be calculated by multiplying the force applied to the outer edge of the section by the perpendicular distance between the force and the axis of rotation.
To find the perpendicular distance, we need to consider the geometry of the rotating door. Looking at the drawing, it seems that the door is divided into four rectangular sections, so the total length of the door would be 4 times the length of one section.
Since the force is applied perpendicular to the section, the distance between the force and the axis of rotation would be half the length of one section.
Therefore, the perpendicular distance is equal to 0.5 times the length of one section.
Substituting the given values, the torque τ = (74 N) * (0.5 * length of one section).
The moment of inertia for a rectangular section rotating about an axis perpendicular to its length can be calculated using the formula I = (1/3) * M * h^2, where M is the mass of the section and h is the length of the section.
Substituting the given values, the moment of inertia I = (1/3) * (70 kg) * (length of one section)^2.
Finally, we can calculate the angular acceleration α by rearranging the torque equation: α = τ / I.
Substituting the calculated values, α = [(74 N) * (0.5 * length of one section)] / [(1/3) * (70 kg) * (length of one section)^2].
Simplifying the expression would give us the magnitude of the door's angular acceleration.